I think he will largely stick to it, mainly because he has the possibility to do things outside the manifesto pledges without claiming he has broken it.
Manifestos aren't binding because circumstances can change. There is probably a claim that a lot has happened from an economic perspective since Starmer was elected.
However I think Burnham has indicated he won't go back on the election pledges, such as not raising income tax.
If he is smart then there are plenty of changes he can make to taxes for the good that aren't mentioned in the manifesto or even are mentioned but have been undercooked by Reeves. Such as CGT, inheritance tax, council tax. Reeves herself proposed changes to council tax but fluffed the opportunity and there a lot more things that could be changed along with these.
I think if he does make changes to manifesto pledges there will be a lot of pressure for an election and probably rightly so. I guess for him there is a question, does he hold an election before or after those changes.
I suspect a lot of things he is going to do will be for the good of the country and will actually be the right thing to do economically. But I also suspect they are going to annoy a lot of the richer people and generate a considerable amount of rage/media pressure.
Whatever changes he does do re tax he needs to focus on selling them hard to the people that benefit from them and explain why they are necessary, rather than doing what everyone else seems to do, suggest a policy and then just U turn when a few people on the internet complain about it. If you are going to do something do it for the right reasons and don't bottle it because the angry media doesn't like it.