My father died a year ago in North Carolina. He had a will and left everything to his wife. He shared ownersip of heir property with his siblings. My mother-in-law does not want to continue to pay taxes and would like to transfer her interests in the properties to me and my 3 siblings. Is a quitclaim deed the best way to accomplish this?

Relinquish = gift in the eyes of the IRS, and gets complicated when real estate is involved.
Quitclaim deeds are often used between family members (but I assume she never adopted any of you so you are not family?), and quitclaim just means that there are no guarantees or warranties about the property, in regard to clear title, taxes owed, anything.
Without answers to a lot of questions, such as what she will be doing for the rest of her life, and how the 4 of you plan to own and be responsible for the property, you need to talk to a lawyer.
In general, gifts of real estate are not a good idea, for her or for you.

Because issues of how it was transferred though the estate ( if it was) and that it involves step children and several family members I would seriously suggest a real estate attorney to do deeds for transferring it, The family members getting the property should be willing to pay for this, and it will make sure all the things are done correctly to solve future issues

First, was the will probated? Has the estate been closed?

Second, the relationships here are confusing. You say; "He shared ownersip of heir property with his siblings". That seems to say your father shared the property with HIS siblings so the wife only got his share. Now she wants to relinquish her share to you and your siblings.

If probate of the estate has not been completed, this is done differently. If it has, then it means she will have to "sell" the property to you and your siblings, with the tax consequences that go with it.

So, essentially we need more and clearer info to be able to help you further.

Your father died.. and your mother-in-law doesn't want to pay taxes on property which she inherited?

What's the connection between the mother in law and the father?