When you ask about untapped kegs, what scenario are you imagining? Are you worried about not having space in a keezer/kegerator for all your kegs, or do you just want to know what to do with them while they're in the keezer/kegerator but before you're ready to drink? The carbonation will be a lot different at room temp than cool. I'd recommend always keeping full kegs cool, unless you're naturally carbonating them. You can't really store beer long-term in a keg without carbonating it, because you need to keep some pressure on it in order to ensure oxygen doesn't get in. Most kegs can't reliably keep a seal without some pressure. And of course when you pressurize it, the beer is going to absorb the co2 and become carbonated. (If you're leaving the keg hooked up to your gas supply, then it will continue to pull in co2 until the beer is carbonated at the level of the set pressure. So you'd want to leave it at serving pressure of 12-15 psi for long-term storage. If you're not leaving them hooked up to your gas supply, you will need to re-pressurize the keg every 12-24 hours until the beer is carbonated, then it will keep the set pressure until you tap it.) You will get better and more easy-to-predict carbonation when your beer is cool, rather than room temp.
Bottom line, you can leave a keg sitting untapped for basically as long as you want if it's hooked up to your gas line and pressurized. If it's not hooked up to your gas line, it's a little more high maintenance until the beer and the headspace reach equilibrium and it can keep pressure. Either way, I'd plan to keep the keg cold.