It's all about the lawyers. I can back up those who say that the odometer is more accurate than the speedo. Most manufacturers make the speedo read about 2 mph high around 70, from all the data I have collected with GPS devices in rental cars over the years. It keeps the automakers out of trouble.

i have considered getting tires that is a size bigger. Will I wear out or damage anything by doing so?

What do you mean by "a size bigger"?
What tire size do you currently have and what size(s)
are you considering?

No "damage" that I am aware of, with "+1" sizing...
some size "ups" may affect your speedo even more than
the std optimism.

A couple of Tire size Comparo Charts:
Tire size calculator

Tire Size Calculator - tire & wheel plus sizing

GL, mD

The trouble with upsizing the tyre is that your hytherto reasonably accurate odometer will now potentially under read. It is better to just understand that your speedo overreads by a few percent and compensate for it when driving.

I agree. Changing tire sizes in order to get the (innacurate) speedo to work will throw off your odomter and OBC (both accurate), so you're not really gaining anything. I know in my M5 I can change the percentages through the OBC for adjusting the MPG averaging. It's a software thing that controls our speedo's as well since they're not old school cables anymore, so there simply has to be a way to change some value in the computer. People are doing the somewhat complex job of coding (myself included) 740iL's into euro 740i's for 6spd conversion, so there must be a screen somewhere to change the accuracy/offset of the speedo.