My startup took some small seed funding with a decent valuation last year. Since then, one of the co-founders has removed himself so we need to reflect that in the operating agreement and transfer some of the equity to his replacement. Does the original co-founder need to sign an MOA (memorandum of understnading) or affidavit to make it legal? How can the new co-founder avoid the tax implications since the company has a valuation so the equity is actually worth something?
Neither an MOA nor an affidavit are required.
If by "operating agreement" you mean that you have an LLC, then you should follow the operating agreement in so far as it dictates how member departures/additions are handled. As for the transfer of equity, there needs to be some contract underlying the transfer of membership interests from the LLC to the new member.
If you're a corporation, then a simple stock purchase agreement will suffice to transfer the shares to the replacement.
I think he's got tax implications right off the bat, unless they don't vest right away, but even in that case he has liability when they do vest.
The company might want to offset that liability - but that also needs to be considered.