I did a bit of poking around on the internet, and found a few message boards that seemed to provide anecdotal evidence that if a veteran was honorably discharged, then they would still be honored at their funerals, even if they committed suicide.
I thought these links might be helpful also (they do not list the cause of death as factors that would make a vet ineligible):
Veterans' Affairs--Burial Benefits and eligibility
Arlington National Cemetery
You could also ask the Dept of Veterans Affairs about this directly--Here's a link to an online form: https://iris.custhelp.com/app/ask/session/L3RpbWUvMTMyMTEyNzQyNy9zaWQvS0ZVMWZZSWs%3D