CPW was quite right, the small texture sizes used in Minecraft mean that even going from 16x to 64x hardly even makes your video card notice. Modern games have much larger textures; for example, the largest texture size a state of the art Radeon 7870 video card can work with is 16,384x.
However, what CAN reduce your FPS is a lack of memory, both on the CPU and on the video card side. Modern games use highly complex and efficient texture compression, but Minecraft's textures are uncompressed. Therefore, using HD textures makes your memory footprint explode - for example, a 64x texture pack will rapidly fill up my GTX 460's 1 GB VRAM and keep it topped up. Java itself also seems to get more memory hungry; at least, when I run a client and a server on my computer (6 GB RAM) at the same time, I get low memory warnings with a HD texture pack and no warnings with default textures.