It won't hurt the guitar any, but if you take all the strings off, it will take a few more minutes tuning it after you put on new ones, as the neck will relax a little without tension, and will bend back a tiny bit slower, causing the need for some more tuning - even on hardtails.
But it's nothing major. Still, I always change strings one at the time, stretch them properly at the same time & tune right away, because that makes for a very quick string change and it'll stay in tune straight away. I always also make sure I use the exact same string gauges every time.
One string at a time is pretty much a must do with Floyd-equipped guitars, though, again so that tuning is much easier. The only time I take 'em all off is once a year when I clean the fretboard properly and check if it needs some fret filing/polish, and even then I usually leave high and low E strings in place, but pulled over the neck sides, because on some guitars the bridge falls off unless it has strings over it.
Dee
"When life's a biatch, be a horny dog"
Amps: Marshall JVM 410H w/ Plexi Cap mod, Choke Mod & Negative Feedback Removal mod, 4x12", Behringer GMX110, Amplitube 3/StealthPedal
Half a dozen custom built/bastardized guitars all with EMG's, mostly 85's, Ibanez Artwood acoustic & Yamaha SGR bass, Epiphone Prophecy SG, Vox Wah, Pitchblack tuner plus assorted pedals, rack gear etc. for home studio use.