Would "OpenSourceRouter" be a trademark violation?

I've built one in Rust and want to market it.

Not if you’re in a different domain, then it cannot be claimed to confuse customers.

So if it’s a network router or a woodworking router, then you’re good.

I can sells of red socks, and white socks, without getting sued by Chicago or Boston.

If it’s in the same space… you might want to contact a lawyer. I believe Firefox had conflicts with earlier names like Phoenix and FireBird and they weren’t even browsers.