Will whitespace (i.e. spaces and tabs) mess up php?
Not directly, no. But it can.
PHP gives you the ability to modify the actual HTTP headers directly. WordPress uses this ability. However, if there is actual output, including whitespace, before the attempt to modify the headers, then you get an error. This is because the headers must come first in the resulting output.
HTML does not give you the ability to modify headers. PHP is generating HTML code as output, but it has those headers to deal with too. So yes, in certain cases, badly placed whitespace can break your page.
Now, it won't do it in (most of) the Template files, which is the only thing an inexperienced user should ever be editing. But it can do it in plugins or the functions.php template file.