Does killing yourself count as an elimination?
Yes and no. Normally, self-damage from explosives and the like does not give the kill to anyone.
RIP-Tire's case is different. It's the same kind of entity as an hero, so kills CAN be attributed to it. This means if you kill Junkrat with the tire, it will count as a kill and it'll even show "Eliminated by [yourname]" along with "[yourname] eliminated".
Fun fact : for some reason, this counts as a shutdown, so there have been some cases of Junkrat players getting PotG by killing only themselves.