My 2 favorite cartridges for varmint are 220 swift and .22-250. Both are fairly rough on barrel throats. ESPECIALLY when I take them on P-Dog shoots. If I'm in a highly populated colony, even switching out rifles to let one cool, I've been known to burn out the throat on my barrels on one hunt.
As Cal put it though, there are many mitigating factors for your choice of varmint weapon. If I'm going to be shooting at yotes past 300-400 yards, I'll break out the 7mm-08 or the .243 that I have set up for longer range shooting. If I'm calling in the hills where my shots wont be past 40 or so yards, I use the 12 gauge and have one of the rifles on standby for that past 40 yards shot that seems to always come because I suck with a shotgun

Sad aint it? I can bang a 16 inch gong at 800 yards all day with a couple of my .243's yet can't hit the broad side of a damn barn from the INSIDE with a shotty!