My Stihl FS85 runs good for about 20 seconds and then shuts completely off. I have replaced the Plug, cleaned out the fuel system and it still does the same thing. Could it be the coil?

cleaning the carb might not clear problem may have to rebuild it also check fuel is correct and look at fuel filter inside tank

STIHL TRIMMER-WEED EATER REPAIR: stihl carburetor ...

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www.youtube.com/watch?v=vK6MsQalprY
Sep 16, 2011 - Uploaded by JEREMY FOUNTAIN Here is what I did for my fussy Stihl FS 250r trimmer....this after sitting all ... STIHL FS80 FS85 FC85 HS80 CARBURETOR ASSY 41371200614 ...

First check your muffler's spark arrestor to see if it is clogged.

If not clogged try adding a little fuel mix introduced into the carburetor throat. If starts then you have carb problems.

If it don't start then you may either a compression problem or ignition problem.

I had the same model and always had a hard time getting it to start after taking it to shop it started three times ok then the hard to start all over again. It now is in the trash as I refuse to put any more money into it. Stihl makes a great chainsaw but I would never buy another trimmer or brushcutter from them again.

Check fuel filter, fuel tank vent and possibly clean carb and replace carb diaphrams

fuel tank vent
bad inline fuel filter
leefs inside the tank flush it out
bad plug replace it
bad air filter replace it
bad pickup ignition coil

oil level too high or low take it all out and measure what you put back by the user manual from the stihl site {use google to find it}

Pumping the primer bubble while the engine is running should cause the engine to over fuel and stall (if the tuning is correct). Sounds like its not fueling enough, pumping the primer is forcing fuel through the carb causing the engine to pick up rev's, but this is soon exhausted which is why it wont rev on the throttle.
* Check that the fuel line is not damaged and drawing in air. (if this is intact try the next step)
* Strip and clean the carb. Replace any diaphrams if needed.
* Check gaskets seal between carb, air filter and cylinder.

Just out of curiosity what standard carb settings did you use.?

change to stihl ultra oil and make sure you ratios are correct. also replace spark plug and spark arrester.

Make sure the carb settings are correct. Screw both tuning screws all the way in and then turn ot anticlockwise the following settings H 3/4 of a turn, L 1 full turn. (this is set to the standard setting)

I suspect carb. trouble. Is the compression OK? Look at the spark plug after trying to start it. If it is wet, you are getting too much fuel. The white smoke indicates that. You may need to overhaul the carb.
Go on the Stihl site and customer service for information.