Here is the thing:
The Constitution does not SPECIFICALLY prohibit placing a tax on religious organizations. (The idea that "free exercise thereof" means no taxes because a tax would be a government induced burden on churches is, IMO, a bit too much of a stretch to work.) However, you will find that the practical application of a tax on religious organizations is impossible without violating the Constitution.
For instance, let's say we tax contributions to religious organizations. Do we tax them at the same rate, or on a progressive scale? Well, we can't tax them on a progressive scale, as that would be placing a heavier burden on larger churches over smaller churches, or larger, more established religions over smaller ones. Sorry, we cannot do that - it would violate the principle of not favoring one religion over another simply because we feel the larger religions are "able to pay more".
So, do we place a flat tax? But what does that do to the small, poor churches who are already struggling to keep their doors open? Well, since we cannot favor one church over another, too bad. Except then the government ends up in the position of shutting churches down as the law of unintended consequences shows its ugly face. So flat tax doesn't work either. Flat tax with a standard deduction - well maybe, except that no matter where we place that deduction, it will still end up harming those borderline churches which are not quite poor enough fall under the standard deduction, but not wealthy enough to pay without having to make cuts in their other expenditures - expenditures which government has ZERO authority is telling them "Well, you don't really NEED that to worship." Flat tax with a variable deduction? Nope - we cannot favor one church over another, no matter WHAT the justification.
The bottom line is there is no way to structure a tax on churches that will not, effectively or directly, end up favoring certain churches over others. Of course, that is a direct violation of the 1st Amendment.
Then we come to the problem of enforcing a tax on religious institutions. What does the government do if a church falls behind in their tax burden? Or, from another possible (probable!) angle, what does government do with a church that refuses to pay? Does government shut the church down and confiscate church property to satisfy the debt? Yea, THAT will fly well. Do they send in law enforcement to forcibly remove offerings until the debt is settled? Sorry, I don't think that one will work, either.
The end result is that while the Constitution does not specify that religious organizations cannot be taxed, the reality is that it effectively does so due to the impossible nature of designing and enforcing a tax structure that will not violate those direct provisions in the Constitution which protect religion from government interference.
