When using popen in php, is there a way to preserve the colored output a program might generate? Is there maybe a way I can tell the shell to print all color escape sequences, instead of resolving them?
That depends on the program you are calling. Usually, if a program supports coloured output, it would ask the OS, "am I running on a terminal?" If yes, then it outputs colour codes. If not, it won't. If you run that program through popen(), then the OS would say "no, you're not running on a terminal" and the program would choose not to output the colour codes (because they would be confusing in the captured output).
Some programs may have an option to force the output of colour codes even if output is not being written to a terminal. However, that is not something you can force externally if the program doesn't already have a way to do it.