Can you exile your own card with Rest in Peace to prevent an opponent from reanimating it?

Rest in Peace doesn't let you choose which cards to exile — instead, it sets up a continuous replacement effect (CR 614.1) that replaces any event where a card would be put into any graveyard with that card being exiled instead. This applies to cards owned by all players equally.

So while you cannot selectively target your own card for exile with Rest in Peace, the effect does automatically protect your cards from reanimation strategies: since your cards never reach the graveyard in the first place, there is nothing there for an opponent to reanimate. The replacement effect applies as the card would be placed into the graveyard, under CR 614.6, before it ever arrives there.

It's worth noting that Rest in Peace also exiles cards already in graveyards when it enters the battlefield (due to its first ability), sweeping all existing graveyard cards into exile. After that, its replacement effect handles all future cards.

Example: Your creature is destroyed and would go to your graveyard. Your opponent controls a Reanimate and is waiting to cast it. Because Rest in Peace is on the battlefield, your creature is exiled instead of going to the graveyard — your opponent's Reanimate has no legal target among your cards and is essentially blanked for this purpose.

So Rest in Peace achieves the goal of preventing reanimation of your cards, but it does so automatically for everyone's cards, not as a targeted or selective choice on your part (CR 614.1c).