Yes, you can play on your new device without having to pay for No Ads again.
On Apple devices:
Open your game and then tap on the gear on the top left of the screen. Then tap "Restore Purchases", this way No Ads should work again.
On Android devices the process is now automatic. Once you start the game on the new device the NoAds feature, you previously purchased, should reactivate automatically on your new device.
IMPORTANT! Please use the same Google Play account or Apple ID that you were using on the previous device.