Why does water constantly flow out of the backwash, even if the filter is on "filter"

No.
Nothing in filter will increase water pressure. Whatever water the pump sucks in will flow out of whatever port is open, in the case of backwash, a Waste port open. So... What the pumps takes in, you will see flow out when you backwash.
No problems that I can think of to look for.

The back pressure at which it's time to backwash a sand filter is about 6-8 psi above the baseline pressure of a clean filter. Just after following the procedure to backwash and rinse your filter, check the sand filter's back pressure at the time you return the pump and filter system back into filter mode. This is the baseline pressure of your clean filter. As the filter gathers particulates from the pool water, the back pressure will rise over time, as indicated on your pressure gauge. Once the pressure reaches 6-8 psi above the baseline, it's time to backwash.

Some advice: Don't assume the baseline pressure will remain the same for your sand filter. The baseline pressure will change over time and when the sand media is replaced. Typicallly baseline pressure will go up slightly over a season due to captured particles that are difficult to backwash out. Use of a filter cleaner will help remove these captured particles.

More advice: Even with the use of filter cleaners and proper backwashing, sand media will not last forever. The constant water flow and movement of particles in the sand over time shears the surface of each little sand grain. Like a river flowing over rocks, over time each sand grain surface is smoothed, and thus its effectiveness for capturing small particles is reduced with ongoing use. A typical rule of thumb is to replace the sand media +/- 5 years. Some pool owners do it more often, some less. It depends on how long your pool season is each year and how much debris the filter much extract from the pool water.

For a pH of 6.0 to 6.9 a type of naturally occurring calcium carbonate media called Calcite is used to neutralize the pH. For water with a pH of less than 6.0, magnesium oxide is blended with Calcite to bring the pH to 7.0 or above. The Calcite or the blend of media is put in either an up-flow neutralizer tank or a down-flow neutralizer tank.

Acid neutralizer water systems are typically installed after the well pressure tank.

In down-flow neutralizer tanks the media flows from the top of the media inside a vertical filter tank down to the bottom of the tank, and up a distributor tube and out of the filter to the household piping. Down-flow neutralizer tanks also act as filters since sediment and other fine particles become trapped in the Calcite. This type of neutralizer is automatically or manually backwashed to keep the media clean.

In an upflow neutralizer the water flows down through the center distributor tube and enters the media bed at the bottom and flows up through the media before exiting the neutralizer and flowing out to the plumbing. Up-flow neutralizers do not get backwashed because the media is never compacted and no sediment is removed. Since the water is flowing up through the media the media is not compacted to theoretically it does not require backwashing.

There are advantages and disadvantages to both up-flow and down-flow neutralizers. The main advantage of the down-flow neutralizer is that it not only neutralizes the water, it also acts as a whole house sediment filter. Down-flow neutralizers are usually automatically backwashed, which cleans the Calcite media and prevents rust particles and other sediment from fouling or coating the media. Since additional Calcite or blended media must be added to most neutralizers once or twice a year, down-flow neutralizers are easier to backwash and put back in service than up-flow neutralizers which cannot be backwashed.

Up-flow neutralizers must use an internal top screen in to order to prevent the Calcite from entering the home plumbing system. Calcite has the appearance of white sand and can quickly damage valves and fixtures if the media enters the plumbing system. If the water contains iron, manganese or sulfides, these internal top screens can later become fouled and so are generally are not used for this reason. Instead of the internal top screen, a filter housing and cartridge filter are usually installed after the up-flow neutralizer tank to prevent any mineral from flowing into the plumbing system. AdChoices
With down-flow neutralizers these upper screens or external filter housings are unnecessary since the Calcite is prevented from leaving the filter tank due to the bottom internal distributor screen. The bottom distributor does not get easily fouled due to the backwashing the down-flow neutralizer tank receives on a regular basis.

In filter tanks the media can flow around the media and create channels which allow the water to flow without properly contacting the media. This type of channeling is more of a problem with up-flow neutralizers and rarely happens with down-flow neutralizers due to the action of the backwash. For most residential applications down-flow neutralizers work better than up-flow neutralizers due to the filtration feature and the backwashing function.

A down-flow neutralizer can be backwashed on a regular basis to clean, re-classify and distribute the calcium media thoroughly. This backwashing allows the down-flow neutralizer to function properly and lower maintenance costs. The Calcite media dissolves better because it is cleaned and then compacted in the down-flow neutralizer tank.

Well water that is acidic can also sometimes be high in iron, manganese or hydrogen sulfide. If a greensand or other type of manganese media iron filter is used to treat the water for iron, the pH should be raised up to at least 6.8 to allow the iron filter media to work properly. A down-flow neutralizer is usually the best choice to put in front of an iron filter because the neutralizer acts as a pre-filter removing some oxidized iron prior to the iron filter. This lessens the load of iron that the greensand filter must treat. Iron, manganese and sulfides can coat the acid neutralizer media and render it unable to dissolve into the water and neutralize the pH.

There are some applications where up-flow Calcite neutralizers are more desirable than down-flow neutralizers. If the flow is more or less constant on a regular basis, such as when the neutralizer is used to fill a holding tank with neutralized water, the up-flow filter works fine. Since up-flow neutralizers have no automatic backwash control valve they are less expensive than down-flow neutralizers. If the water is of excellent quality with no sediment or iron and the flow rate is constant then the up-flow neutralizer costs less to use and uses no backwash water.

When the filter get clogged the pressure in the filter increases and there is less water flow returning to the pool.
If your filter is a sand filter, when you turn the valve to backwash and turn on the pump it reverses the water flow in the sand filter to flush the dirt out of the filter to waste. Look at the window openning in the waste pipe and keep the backwashing goinging until you see clear water.
Then turn off the pump and move the control valve to Rinse, turn on the pump for a few minutes, this returns the water flow back to normal but the water goes to waste to make sure the sand filter is clean.
Turn off the pump and move the control valve to filter and turn on the pump. This mode is the normal operating mode and the pressure gauge in the sand filter will show a drop in pressure.
Some pressure gauge shows a red section to indicate when backwashing should be carried out.

FILTER is the valve position when the system is in the normal filtering mode.
When you backwash the filter you first need to turn off the pump, place in BACKWASH position, and run the motor for 45 seconds (or until the backwash water is clear). Turn off pump and run in the FILTER position for 15 seconds. Repeat twice more.
The RINSE position is to clear any dirt in initial start-up cleaning, plus resetting filter bed after backwashing.
The WASTE position is for vacuuming directly to waste, lowering pool level, or draining the pool.
The CLOSED position is for shutting off all flow through the valve.
The RECIRCULATE position allows water to flow from the pump through the valve and directly to the pool.

You will probably only to use the FILTER, BACKWASH and maybe the RINSE valve position. The RINSE position is not as important as indicated by the fact that many backwash valves don't have a RINSE position.

I hope this helps.

b> VALVE SETTING FUNCTIONS OF VALVE POSITIONS FLOW DIRECTION THROUGH VALVE FILTER PUMP - TOP - THROUGH FILTER - BOTTOM - RETURN For normal filtration and vacuuming pool through filter. BACKWASH b> PUMP - BOTTOM - THROUGH FILTER - TOP - WASTE For reversing flow for cleaning filter. RINSE b> PUMP - TOP - THROUGH FILTER - BOTTOM - WASTE For initial start-up cleaning, plus resetting filter bed after backwashing. WASTE b> PUMP - WASTE For vacuuming directly to waste, lowering pool level/draining pool. CLOSED b> NO CIRCULATION PAST PUMP PORT For shutting off all flow to filter and pool. RECIRCULATE b> PUMP - RETURN For bypassing filter, but circulating pool water. May be plumbed for "off-system" pool water access. Ideal for Jet-Air® fittings. WINTER b> VALVE NOT IN USE
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When cleaning pool filters that use cartridges, there is never going to be a backwash option. The cartridges can be cleaned and reused, but you have to take them out of the unit in order to clean them.

1st Green water is algee. Kill the algee in the pool. 2nd when you're finished backwashing the filter. Stop pump and move valve handle to rinse position. Then restart pump. Run in rinse mode until glass is clear. Then stop pump again, and move lever to filter position. When you're backwashing, you're reversing the flow of the water through the filter. When you switch directly back to filter, anything left loose in the filter flows back to the pool. When you select rinse, it reverses the flow back to normal, but keeps the water going to the waste outlet until clear, that way that green puff of water will go out the drain. You can get complete manuals from www.haywardnet.com

Check the Filters and Skimmer FIRST even though they said they backwashed and checked the system. I am not sure how you determined it is the water pressure switch? If you do have sufficent flow of water in and out of the heater, the pressure switch could need replaced or the small copper feed line from the heat exchanger to the pressure switch could be obstructed. You really need a Heater Tech that can check the pressure switch for proper functioning and any obstructions in the feeder line, Normally you will find the Skimmer and or Filters are the cause of low water pressure.

For the heater to burn, there must be water pressure and flow - and the heater must know of it. It is not enough that you know the pump is on, the heater must know it. Check your heater's water pressure switch.

What about the filter pressure ... high filter pressure would indicate need for a backwash. dirty filters slow the water flow. If your filter normally runs at 20 PSI (when clean) it should be backwashed at 30 PSI, rule of thumb = 10 psi over 'startup pressure'.

Your heaters job is to heat the water ... think of it as a very agressive bully. The water is absolutely necessary to keep the bully in check.