Are horses slower with armor?

In minecraft armour does not affect the player or a mob's speed, the diffrence in speed is because of the random NBT tag the horse has for the speed.

Compared to the player's walking speed, the maximum speed ranges from merely 25% faster to 337.5% times faster.

All nbt values can be found here (gamepedia)

I have played many hours of Minecraft and I have never noticed my horse go slower. So if it does, it is so minuscule that the armor would be worth it.

Also Google your question; there are several videos testing this, including a Mythbusting episode by Xisumavoid:

The armor test starts at 1:20.