In PttM the campaign card instructs you to remove the burdens from the Treason of Saruman, are you supposed to shuffle the burdens from the Black Riders (the same ones that were removed during Treason of Saruman) back into the deck at this point? Or are those burdens removed forever during Treason of Saruman?

Yes, you're supposed to add the Black Riders burdens in. They weren't removed from the campaign pool, just excluded from the three ToS quests (presumably because they're mostly to do with the Ring-bearer, who isn't there for ToS).

I thought that if campaign card instruct remove burdens, they are remove permanently.

I thought that if campaign card instruct remove burdens, they are remove permanently.

Only if it tells you to remove them from the Campaign Pool. If it only tells you to remove them from the encounter deck, that means just for that quest. From the Uruk-Hai campaign card:

Remove Gildor Inglorion and Mr. Underhill from the Campaign Pool. Remove each burder with the following burden set icons from the encounter deck

Observe the difference: the boons are removed permanently, the burdens are not. Note that a particularly sneaky rules lawyer could argue that since this setup card doesn't tell you to remove Gildor from the encounter deck, he is available to help out in the Uruk-Hai, but this doesn't seem intended behaviour.

As further proof that the removed burdens come back, observe that the Helm's Deep and Road to Isengard campaign cards also tell you to remove these burdens. If they had been removed from the Campaign Pool, then these instructions would not be needed because they'd already have been removed. Further, Passage of the Marshes instructs you to remove burdens from Road To Isengard---if this removed them permanently, they'd have have a chance to be used!