What will happen if I restore from an old iPhone backup with a different Apple ID?

I suggest you sign into the old ID on the new phone and restore your backup to the new phone. If you were syncing data with iCloud, set up iCloud syncing, and let that finish. The photos sync is extremely slow. It could take several days or more for the photos to sync.

Once the above is done, when changing to the new ID, there may be some data loss, but you can minimize it. If syncing photos with iCloud, ensure Optimization is turned off in Settings/Your Name/iCloud/Photos. Then make sure full resolution photos are on the device. When signing out of iCloud, if asked whether you want to retain data, choose yes. Messages conversations will be lost if they are addressed to the ID. See the article below for other places that the old ID should be removed. See Winston Churchill's response in the below discussion. If possible, also set up Family Sharing  between the IDs to keep access to your apps/purchases. 

 How to switch Apple IDs on your iPhone or iPad.      

How to transfer all files from one iCloud account to another.       

What to do after you change your Apple ID or password - Apple Support.      

Family Sharing.         

Family Sharing - Set up.        

You might restore some data you won't be able to access and purchases you won't be able to update because they are tied to the old ID. Why are you changing Apple IDs? If you stick with the old ID, you retain access to everything associated with that ID.

I am guessing you are saying you have a new phone with a new Apple ID, I would say it would keep your apps and 3rd party data, but you may not be able to access some subscriptions, make sure to cancel all subscriptions and update websites with your new email.

Your situation is similar to the one tackled in this link. I believe, Eric Root also responded to the inquiry posted on the link I provided.

Regards,

RRB