As an example, can you change a time signature from standard 4/4 time to waltz 3/4 and can you change a waltz to standard 4/4 time? Or more generally, can ANY time signature of a song be changed to any other time signature (even unusual ones like 7/4, 9/8) and not completely ruin the song?
I believe you're asking a different question from the one Todd answered. Yes, often. A simple example would be 'Fly Me to the Moon'. Written in 3/4, most commonly played in 4/4. Sometimes for fun, I play 4/4 numbers in 5/4. 'Lullaby of Birdland' is a goody. Care needs to be taken with the phrasing and timing (obviously!) but a lot of numbers can be changed about.
It does change some drastically, especially vocals, but it can be and is done. But not all songs will benefit...
There's a version of Take Five that's been recorded in 4/4. Somewhat destroys the point...
The time signature can vary throughout a song, but it sounds like you are asking if, for example, you can play a 4/4 song in 3/4. That depends on how compatible the signatures are and what you consider "completely ruining the song."
4/4 and 2/2 are interchangeable. 6/8 and 3/4 are usually interchangeable. 12/8 can be compatible with 6/8 and 4/4 swing. Beyond that, you'll likely have to make some adjustments (e.g., to play a 3/4 song in 4/4, you will have to add 1 beat to every measure), and those adjustments could make the song very awkward.
The beatles "with a little help from my friends" is in 4/4 and joe cocker's version is in 12/8. I think both were hits, if that is any indication of not ruining a song. The song is wildly different in style so how much of a change does one need to make before it is considered a different song is up to interpretation that i will leave up to you (and the copyright lawyers).
Listen to the flecktones play the 12 days of Christmas for a definite "yes". Each day is a different key and time signature.
In addition, it's possible to change time signatures to whatever you want, so long as you make the apporpriate use of tempo changes and tuplets. Basically, you can change the time signature to whatever you want, and it's possible to undo its effect by making notation that makes people's eyes bleed. If you want to change the feel of the music (change relative note durations), yes it's possible, for example "What a Wonderful World" done by Louis Armstrong is in 12/8, but commonly the song is played in 4/4, and it just feels like all the swing has been traded for a faster feel.