Do followers lose items when replaced with companions?

They don't lose the equipment you give them, but they will revert to their old loadout. You will need to give or take a piece of armor or a weapon for them to equip the best things they have.

Also, watch out for invisible companion gear duplication, sometimes it happens after dismissing and rehiring a follower, leading to their decreased carry weight capacity. You may be able to pickpocket the duplicates after dismissing a follower, if they are unequipped, or, if you're on PC, remove them via the console.

When I finally purchased my first house in Whiterun my previous companion, Lydia, was there with her items still equipped.