Armors do not have a fixed maximum speed. Instead, medium and heavy armors reduce your speed. A typical human in normal full plate has a move of 20, but a typical halfling in full plate only has a move of 15. In one case, 30 is decreased to 20 and in the other, 20 is decreased to 15.
Assuming the effect does not say anything about heavy/medium load or armor, it will increase your speed, which will then be decreased by the armor. A hasted human would have a speed of 60, which is reduced by the armor to 40. A hasted halfling would have a speed of 40 (haste limits to max twice the base speed), which is reduced to 30 by the armor.
A human barbarian in full plate would only move at 20, since barbarian fast movement does not apply when wearing heavy armor.
There is a table (possibly only in the PHB or DMG since I cannot find it in the SRD) that gives the resulting speed for base speeds other than 20 or 30.
I found the table. It was hidden under carrying capacity at
Carrying Capacity :: d20srd.org. Roughly, take the multiple of 5 that is between 2/3 and 3/4 of the base speed.
Mithral will reduce the category of an armor by one level, so you could move at full speed in a mithral breastplate or run at x4 in mithral full plate. I'm not aware of other means, although I'm certain there are at least some properties in some of the expansion books. MIC might also have an enhancement to do that.