Seeing as you dont provide no link or list any useful info, like hacks installed, ill ponder a guess.
When they try to login, they login at
www.ursite.com, when they automatically get logged out, they are at ursite.com, note the missing www.
You running vbadvanced?
Change your cookie domain in the admincp to .ursite.com, note the dot on the front there.
Tell the members having issues logging in, to delete their cookies from their browser and restart their browser and then try logging back in.