I noticed that any time I install Ubuntu 11.10 or 12.04 alongside my Windows 7 OS, the Windows partition runs very slowly and the hard drive always seems busy in Windows 7. I have had a hard drive completely fail in the past -- could the dual booting be the problem?
No, dual booting can not cause any harm to your harddisk. It only stores the beginning and end of partitions which each contains one filesystem. The partitions simply says at which byte of the disk the filesystem starts and ends. The operating system will then only work within that area. So if your partition table gets messed up, you can get some very interesting results. It could seriously damage your data, but your harddisk wouldn't care at all. To the harddisk, each byte is as good as another.
Particularly in Windows, it is quite common for harddisks to seem slower over time. It's not because the disk becomes slower, but because the data gets scattered around, meaning the harddisk has to do more work to get the same job done. You can defragment your disk in Windows using its built-in tools or third-party tools, like Perfect Disk.
One thing you should be aware of, is that this happens more quickly if your filesystem is very full. If it's an option, you should always try to have at least 20% free space. So if, while installing Ubuntu, you resized your Windows partition that left very little free space, this can explain why it slows down.
Other than that, two filesystems do not affect one another, and the disk doesn't care about filesystems at all.