If the partnership has income, that income, and the tax obligations, are passed on to the partners via the K-1 statement. Someone has to pay the tax.
Broadly speaking, if you invested $100,000 in the partnership and you are bought out for $120,000, you have $20,000 of taxable profit. The entire proceeds are not taxable. If you were bought out for $90,000, you would have a $10,000 loss, which might or might not be deductible on your regular tax return, depending on the nature of the partnership.
Where are you are running into tricky territory is that you are trying to do two separate things as if they are one thing. Right now, the partnership owns a house. You want the partnership to sell that house. By itself, that is a straightforward transaction. The partnership would divide the profits between the partners and the partners each pay the tax on their share. You also want the other partner to buy out your interest in the partnership. By itself, this is also fairly straightforward, assuming that the partners can agree on a price. It gets much trickier when you’re trying to combine those two things in a single transaction.
