Why doesn't the treaty reduce my US tax as a citizen?
Because most treaties contain a saving clause. It preserves each country's right to tax its own citizens and residents as though the treaty were not there. So when you read an article that appears to exempt a category of income, you have to read it again against the saving clause: for a US citizen the article often does nothing, because the United States has kept the right to tax you on the same basis as any other citizen. The treaty still does real work on the other side of the border, and a short list of articles is usually carved out of the saving clause. Which ones are carved out is a question about the specific treaty text, not a general rule.
What is a saving clause in a tax treaty?
It is a provision by which each country keeps its domestic taxing rights over its own citizens and residents, notwithstanding the rest of the agreement. The effect is structural rather than technical. A treaty is usually read as a set of articles that allocate income between two states; the saving clause sits behind those articles and, for that country's own people, switches much of the allocation off. It is the reason two people with identical income can get opposite answers from the same treaty, one being a citizen of the taxing state and the other not. When a treaty position is being claimed, the saving clause is the first thing to read rather than the last.
Are there exceptions to the saving clause?
Yes. Treaties containing a saving clause generally follow it immediately with a list of articles the clause does not touch. Those exceptions are the only places where the treaty does what it appears to do for a citizen of the taxing state. They vary between treaties and between versions of the same treaty, and some are narrowed further, so an exception available to a person who is not a citizen or a long-term resident is smaller than it looks on a first reading. The practical step is to find the exception list in the text actually in force, check whether your article is in it, and record where you looked.
Does the treaty tie-breaker stop me filing a US return?
No. The residency article can decide which country treats you as resident for treaty purposes, and that matters a great deal for how income is allocated. It does not remove citizenship, and the saving clause preserves the United States' right to tax its citizens wherever the tie-breaker lands. In practice you can be treaty-resident in one country and still carry a full citizen's filing obligation in the other, with relief coming from foreign tax credits and from whichever articles survive the saving clause, rather than from the tie-breaker itself. Filing positions that treat the tie-breaker as an exit are the ones that attract questions later.
Does the saving clause apply to green card holders?
It is written to preserve a country's taxing rights over its citizens and its residents, so it is not limited to citizenship. A lawful permanent resident is generally a resident for domestic tax purposes, and that is enough to bring the clause into play. Some exception lists go further and withhold the benefit of an article from someone who has held that status for a long period, which is a separate test again. The order of the analysis is: establish your status under domestic law, then read the saving clause, then read its exception list, then read the article you were interested in. Doing it in the opposite order produces confident and wrong answers.
Which country's saving clause applies to me?
Both, in the sense that the clause is reciprocal: each state preserves its rights over its own citizens and residents. Which one bites depends on your status. The clause in the hands of the country you are a citizen of is the one that usually disappoints, because that is the country whose domestic tax you were hoping the treaty would reduce. The clause in the hands of the other country is rarely the problem, since you are not its citizen and its residence test is something you can often plan around. Establishing your status in each country, in writing, is therefore the work that determines what the treaty can do for you.
Which country do I pay tax to first?
Generally the source country — where the income arises — taxes first, often by withholding before you receive it. Your country of residence then taxes the same income and credits what the source country took. That order is why timing matters: a residence-country return filed before the source-country tax is settled has nothing to credit yet. Getting the sequence right is most of the work. See international tax planning.
Do NRIs pay tax on money sent to India?
Sending your own funds to India is a transfer of capital, not income, so the remittance itself is not taxed. What is taxable is income the money then earns in India — interest, rent, capital gains — under the rules for the account type it sits in. Sending money out of India is the direction that needs certification before the bank will act. See NRE, NRO and FCNR accounts.