It's complicated and depends on a lot of things. On a typical Linux system, bitblt (well, XCopyArea) and font rendering will go through XRender, which may or may not use the GPU depending on driver support. On xf86-video-intel, it will use SNA, which will use the 2D blitter on Intel chips. It's unmaintained at this point and a bit buggy so some distros turn it off. Instead, one might use glamor, which implements XRender on top of OpenGL, which will use the 3D graphics hardware on Intel chips. So it depends on your system configuration.

For other operating systems, the words and details change, but the overall idea of "it's complicated and depends on your driver and OS version and which apps are running and phase of the moon" still applies.