Does having more data on a hard drive slow it down?

The more free space the better it will perform but it may not be very noticeable since Windows makes a reference note as to where everything is stored on the HDD , so it knows where to look for it the thing that is slow is the spinning platters and the read arm.
A drive itself will not slow down mechanically, it still rotates at it's proper speed.

The outer rings of the drive are longer, and will contain more data. When you read or write from them, you will get more data per revolution, thus a higher transfer rate.
The OS starts loading and using the outer rings first. That is why a new drive seems to be a bit faster.

In terms of seek or access times, a full drive will cause the access arm to have to traverse a longer distance than when ther is data on only a few cylinders of a new drive.

Also, a fullish drive may not have contiguous free space available for new files, and will have to put pars in different locations. It helps to defrag drives that are fullish.

If you really want performance, get a SSD. You are looking at 50x speed in random i/o and 2x speed in sequential.