Has the 5 LED blink. Are there any tests I can run to narrow down how to go about fixing this issue. False flame I think it is.

Sounds like a bad board. If this is electric heat you can check continuity across the limit switches with a multi meter. Turn power of to unit and remove wires on each side of limit to test. Really could be a number of things. Good Luck.

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If you narrowed down, then replace the power board. What else. These blinks [flashes] are fault indication (Error Codes). From the number of blinks occur at one time switch ON, you can get an idea about the faulty circuit section inside it. This will make repair easy. These are called blinking codes. Make sure about the exact number of blinking of the standby/power LED, by switching On & OFF the set, for at least two times. Count it carefully. If you wish to get some details, check the site linked here. It has details about blinking codes, and fault locating procedure, to many popular brand TVs and other devices.
http://electronicshelponline.blogspot.com/
http://electro-medical.blogspot.com/
Blinking codes details of many popular brand TVs are given. Surf the site with patience.

62-24084-02 LED CODES
POWER LED - GREEN
ON - Normal
OFF - No power to board
OK LED - GREEN
One Blink - One hour retry delay
Two Blinks - Pressure switch failure (remains open)
Three Blinks - Limit switch fault
Four Blinks - Pressure switch failure (remains closed)
Five Blinks - Twinning fault
FLAME LED - YELLOW
ON - Good flame
Rapid Blink - Flame present with gas valve de-energizing
Slow Blink - Marginal flame

The test light will show power going in or coming out. However the common problem with those module is heat. They tend to break down under load. You can take the module to your local Advance Auto Parts and have it tested for free. I use them often and their tester heats the module.

You must have a bad board.

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besides the beeps which indicate a motherboard timing issue. There should be LED blinking codes given on the front LEDs. I will include the blink codes below, but lets try to at least get the system up and running. First, I would like you to remove the RAM sticks from the laptop and just reseat one stick back in the machine. Try to boot with that one stick, if it does not, try the other alone. There have been issues when reseating the ram or re-installing it have temp. fixed the issue. If it does reboot, go to the HP support site, put in your model number of your laptop and download and install an updated BIOS. Below are the blink codes that may narrow down the problem.....

HP DV6000 LED Blink Codes

LED Component Error Condition
Blinking battery power LED Battery Insufficient charge on the battery
Blink: 1 - Caps Lock/Num Lock CPU CPU not functional
Blink: 2 - Caps Lock/Num Lock BIOS BIOS corruption failure
Blink: 3 - Caps Lock/Num Lock Memory Module error not functional
Blink: 4 - Caps Lock/Num Lock Graphics Graphics controller not functional
Blink: 5 - Caps Lock/Num Lock System board General system board failure
Blink: 6 - Caps Lock/Num Lock BIOS BIOS authentication failure

The board is bad. The unit thinks a flame is present, so it tells the blower to come on. Replace the board.

Go to the set and when turning it on doesn't work push and hold down the MENU and INPUT buttons at the same time until the LED starts blinking. It should be a two blink sequence like 1 blink pause 2 blinks long pause and then repeats. Repost the blink sequence and that will narrow down what the microprocessor thinks is the problem area.

With both of error codes flashing, the problem is trying to figure out wich one to focus on. The flashing 6 code is an indication of HV failure which would then cause a vertical failure(4 blink code). There are so many things that would cause this problem. I can narrow it down to the "D" board to fix your problem. If I were there to troubleshoot it, I could narrow it down further I.E. flyback, HV distribution block. I suggest getting a tech diagnose the exact problem. I wish I could narrow it down better for you. Please take time to rate this solution and if I can be of further assistance, let me know