I have a Stihl FS61 string trimmer that I found recently for free on the side of the road. I tried to start it when I got home, but it only started when gas was poured in the carb, and it only ran for a couple seconds. I purchased a new carb, but still the exact same problem. I did a compression test which read about 170 PSI. I also made sure that gas is flowing to the carb. If anyone has ideas on how to get this thing running, it would be much appreciated. Thanks!

I assume it's an FS 61 not a KS61?
 
https://www.diyspareparts.com/parts/stihl/diagrams/fs-61/  
 
I'd pull the muffler screen (spark arrester) and check that it's NOT CLOGGED with CARBON.
 
If that's ok, try another known good spark plug. While speaking of spark, you may want to pull the flywheel and check that the key is still intact and timed correctly with the crankshaft.
 
I attached a link for your machine.  Stihl diagrams are hard to come by so save it for later use..
 
Pz post back how it goes..
 
 
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Yes, FS not KS, my bad. Thanks for the diagram! I pulled the muffler, but I was unable to locate a spark arrester. I think the previous owner may have removed it. I also inspected the flywheel for damage, but there was none. Should I try to start it without the muffler? Also, I doubt it would be an ignition problem because the engine ran smoothly for the couple seconds I got it to run. I also realized I was running 50:1 gas in it, not 32:1 like it is made for, but I doubt that was the problem.

You can start it W/O the muffler.  If using 2 stroke oil specific for the machine (not marine oil)
you can do 40:1.  The mix your using, as you stated, isn't the issue.
 
The spark arrester is just a metal screen (which sounds like it's already removed)
 
Put a couple teaspoons of 2 stroke oil down the cylinder before starting again.
This helps for a better seal with the rings above AND below the piston...
 
Also inspect around the cylinder to block area for any oil residue, etc. 
You could have a loose/ leaking cylinder to block..
 
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