In theory, yes.
For the server, if it has to send out a 1MB file to each client, it has to spend n amount of time and resources sending out that one file. Now, if you were able to cut the file size in half, the time and resources it would take per user on the server would be .5n.
For the client, it has to download a file. Assuming a download rate of 25KB/S, a 1MB file would take 41 seconds to download. A .5MB file would take 20.5s. Thats a savings of 20 seconds by reducing the file size.
However, in practice. No, I would not worry about it, unless you're dealing with audio/video/picture data. That's because a character in a HTML document is only a couple bytes. Sure, you might have lets say 100 extra characters that you could trim and remove - whitespace for instance. At most you'd save up an additional 1KB per page.
I wouldn't be too concerned about it, unless you're developing an application or solution where it needs to be compact. But any modern or sub-modern computer won't break with 1KB extra data in their HTML file.