The concept of the Red Star on Bucky's arm doesn't exactly have any kind of significant "backstory"... other than that he is the "Winter Soldier", suggesting himself as a Russian operative. This is similar to why Deadpool wears red ("So the bad guys don't see me bleed") or why Batman dresses as a bat ("I must become what I fear"). It's merely a symbol of what his character is, so removing it would pretty much just make him "That dude with the metal arm" (and in my opinion, "Winter Soldier" has a better ring to it).
As for any in-universe reason as to why he doesn't remove it, without any input from the writers, there's very little we can use. That said, there are a few things we can use to ascertain some form of reasoning.
Bucky's state of mind. Bucky has been brainwashed, and as we saw in the movies, how easily it was for him to fall back in line. With a state of mind as fragile as his own, he would need something to focus on, in order to keep himself grounded when his mind starts to slip. You might say the arm itself would act as that anchor, but when you've lived for 60+ years with such an appendage, you'd be hard put imagining what it would be like without it. The star, on the other hand, could work as that anchor.
It is a reminder. Once free of his mind control, Bucky regrets what he did while under mind control. He knows it wasn't his will that made him do it, but now that he has control of himself, he wants to change what the star stands for. It's like a scar - a bad reminder that you can take a lesson from, and make sure you make a better choice.
The only other option is a little less exciting - the star is painted on with special paint, staining the metal itself. Removing it would likely require filing or even the use of acid, and doing that could compromise the whole arm... which nobody wants.