Left, I've studied some tax law. I don't have access to my school's databases yet to look up the tax exempt status of religious organizations. However, I did look up an IRS publication on this matter. Mind you, this is the IRS's interpretation of tax law, Treasury regulations, & court decisions.

According to the IRS, to qualify for tax-exempt status, the organization must be:
*organized and operated exclusively for religious, educational, scientific, or othercharitable purposes,
*net earnings may not inure to the benefit of any
private individual or shareholder,
*no substantial part of its activity may be attempting
to influence legislation,
*the organization may not intervene in political
campaigns, and
*the organizations purposes and activities may not
be illegal or violate fundamental public policy.

According to section 501(c)(3) of the Internal Revenue Code, organization may jeopardize their tax-exempt status if they do not abide by certain rules. These rules include:
*their net earnings may not inure to any private
shareholder or individual,
*they must not provide a substantial benefit to private
interests,
*they must not devote a substantial part of their
activities to attempting to influence legislation,
*they must not participate in, or intervene in, any
political campaign on behalf of (or in opposition to)
any candidate for public office, and
*the organizations purposes and activities may not
be illegal or violate fundamental public policy.

The only thing I can think of is if some other organization challenges the church's status based on the fact the it's "activities are illegal or violate fundamental public policy." However, I'm not sure the picketing is illegal. Like you said,
it may fall under the perview of the 1st amendment. Perhaps Phelps may be guilt of having the church's net holding inured for his benefit.
(I have heard of Christian organizations having their tax-exempt status challenged due of lobbying & political campaigning.)

I hope this helped.