My Teac R-H500 tape deck will not play tapes

It is possible that the brakes are not glued on the bands anymore causing the friction. I would have to open it up[ to see what the real cause is but that would be a god first guess. I have worked on tape decks and at Teac as well as Sony over the years so I am well experienced. The deck probably needs a good going over.
Skywave Tape Deck Repair in Chicago if you need my help

I think I got mine from owner-manuals.com for about $5.00

http://www.owner-manuals.com/index.php?main_page=index

It all depends on where the squeak is coming from. Some very clean rubber parts make kind of a squeak but I don't think that is your problem if you use cartridge cleaning systems. The real answer as a 40 year plus Tape Deck Technician is to get in there and see what is causing the squeak- it may need some oil in reel table, Pinch Roller or even the Capstan Bearing. Manufacturers do not like to put oil into
unit as it makes for no repairs and you will not have to buy a new one.
I worked at Teac but this unit is far after my exit- I still do repairs full time on the product though. I am sure I could resolve it- I am in the Chicago are.

I have not seen those to a large extent due to their value not being worth a repair. I did work on them at Teac though. The capstan motor of the right side deck sound like it is burned out. I have some motors here so it is just a matter of if you want me to work on it or not. I get a lot of units in all the time- I am Skywave Tape Deck Repair in Chicago area.

Replade the capstan belt you can buy it @ partstore.com

In short, no. if you look closely you will see the Teac 3340 has a much larger/thicker capstan shaft than other Teac reel decks. This feature allows the faster tape speeds...7 1/2 and 15 ips. In order to slow down the Teac 3340 you would have to change the flywheel to one from another deck that was made to run at 3 3/4 and 7 1/2. Which is possible but some work..plus you would have to find a slower speed flywheel from another Teac deck. Also to maintain performance, the bias and Eq positions for the slower speeds would have to be readjusted from 7 1/2 and 15.

Could be a worn out belt? The cassette loads it the drive down too much. Hence why it turns with no tape.

very likely that the drive belt has broken. I have fixed a TEAC myself using a large O-Ring from a hardware store (I know I was lucky to find a very close match). Open the unit and find the broken belt and take it to a hardware store to "size up" a replacement. Use the opportunity to oil the gear shafts and clean the rubber pinch roller.