Pretty simple for me.
I agree with their core principles of fiscal responsibility, constitutionally limited government, and somewhat with free market economics.
The first two principles speak for themselves but free market economics is not so simple in my opinion. In a true capitalist society, if a bank used poor judgment and gave loans to anyone who had a pulse who walked through their doors, this behavior would be corrected when/if people defaulted on their loans. Unfortunately we don't allow capitalism to work the way it should and bail out too large to fail banks and insurance companies when they follow poor business practices. It's fair to say that many of these banks were pressured to make many of these mortgages available to undeserving customers so this is admittedly oversimplified but you get my point. Bottom line to me, if your service is so critical to the economic survival of the U.S. that a tax payer bailout may become necessary, then more government regulation and oversight is necessary. If your business is one that can simply fall to the wayside if it fails, then leave it alone for the most part (OSHA, food inspections, etc notwithstanding).
In spite of this, I was on board with their platform and subscribed to their newsletter (which I still receive) but they quickly lost my support for a few reasons:
1) They take Taxed Enough Already too seriously. There are three ways to earn more than you spend (in my opinion): lower spending, raise tax revenue, and stimulate the economy therefore raising tax revenue without raising the tax rate. Without going too far into my personal economic theory, all three have their limits and there are times where taxes need to be raised. Obviously if you go too far with raising taxes, you can kill jobs and create a financial burden on the average citizen but there are times when it's appropriate. I think their "never ever ever raise taxes" mantra is naive and a recipe for disaster.
2) They started venturing outside the scope of their charter. Once I started receiving emails from them in opposition to gay marriage, I completely gave up on them. The economy is enough to keep them busy for several generations; there's absolutely no reason for them to take on outside issues like this and alienate people from what they are trying to accomplish. I personally believe that gay marriage is guaranteed under the 14th Amendment and feel that they are trying to apply the constitution as they find convenient.
There's more than that for me but these are the major points.
Edit to add: I pondered the other day that admitting that you used to support the Tea Party is like admitting that you've been to a Nickleback concert. You permanently lose street cred after that.