You can calculate the size of the effect from simple geometry. Tire wear reduces the diameter of the tire, which reduces the circumference of the tire.
New passenger car tires typically come with 10/32" to 11/32" of tread depth (source). Tires are considered fully worn-out when only 2/32" of tread remains. So the tire has lost 8/32"--9/32" on the radius, or 1/2"--9/16" on the diameter over the life of the tire.
What does this mean in terms of a "percent error?" To find out, we need to know the starting circumference of the new tire.
First, a primer on tire size notation. An example tire size would be 205/45-R17.
- First number: section width in mm. The tire is 205 mm wide
- Second number: aspect ratio. The sidewall height of the tire is 0.45 times the section width
- Third number: rim size in inches. The tire fits a 17" wheel.
Tire diameter is (roughly) given by the wheel rim size plus two times the sidewall height. In the case of our 205/45-R17 tire, the diameter is roughly 616mm.
(17 * 25.4) + 2 * (205 * 0.45) = 616.3
(must multiply the rim size by 25.4 to convert inches to millimeters)
The diameter of this new tire is therefore approximately 1936mm
616.3 * 3.14 = 1936
Now we said we could lose 1/2" in diameter over the life of the tire, so the worn diameter would be 604mm
616.3 - 1/2 * 25.4 = 603.6
Which means the worn circumference is 1896mm, or 2% less than the new tire diameter.
So for this example, over the life of your tire, you'd pick up a 2% speedometer error. Hardly worth worrying about.
Changing tire/wheel sizes can easily have a much larger effect, and care should be taken to try to keep the rolling diameter roughly the same as the recommended tire size.
For what it's worth, my experience is that vehicle manufacturers usually calibrate their speedometers to read fast anyway (they might have legal problems if their speedometers read slow, "I got this ticket for 60 with my cruise control set on 55"), so you're probably not starting from "the truth" anyway. Tire wear would make the speedo read faster still.
A final disclaimer: all the diameter and circumference calculations are ROUGH, as the tire deforms, stretches and slips as it rolls down the road, but I think this is nonetheless a good estimate for the effect of tire wear.