Yes, the Indian constitution does not debar a foreign-born naturalised Indian citizen from becoming the Prime Minister of India. And Mrs. Sonia Gandhi, an Italian born citizen who became an Indian citizen, could have become the Prime Minister of India, if she so desired.
The reason why the Indian constitution does not bar a foreign-born naturalised Indian citizen from this post, unlike the US, is because India's political system is designed around the consensus based Westminster system that offer better safeguards:
The Indian Prime Minister is not directly elected by the people of India and it is the Members of Parliament that elect the Prime Minister from among themselves.
The Prime Minister of India cannot unilaterally make decisions and the source of all Executive decisions is the Council of Ministers (the Cabinet) to which the PM also belongs.
Practically, the many hurdles they need to cross (while also being politically vetted) to become the Prime Minister of India include:
Becoming the leader of a political party (win the support of a large number of dedicated party workers).
Getting elected as Member of Parliament (win the support of a large number of Indian citizens).
Winning support from the majority of the MPs in Parliament.
Earning enough trust to get continuous support of the Council of Ministers and the Parliament to govern the country.
Thus, these checks and balances provide enough protection to ensure that a foreign-born naturalised citizen, with nefarious intentions against the Indian state, cannot rise up the political ladder easily.
These political safeguards ensures that every citizen of India enjoys the same democratic rights, without any prejudice.
As for the second part of the question, "office of profit" means any position for which the government of India (or the government of any Indian state) pays any remuneration and / or bestows administrative and executive powers.
Article 102(1)(a) of the Indian constitution states the rules under under which a Member of Parliament can be disqualified, and holding an "office of profit" is one of them:
(1) A person shall be disqualified for being chosen as, and for being, a member of either House of Parliament—
(a) if he holds any office of profit under the Government of India or the Government of any State, other than an office declared by Parliament by law not to disqualify its holder; ...
(Article 191 (1) of the Indian constitution places similar restrictions on Members of Legislative Assembly of all Indian states).
So yes, anyone with any government job - like a teacher or an IAS officer - cannot be a Member of Parliament or Member of Legislative Assembly unless they resign from their position. The idea behind this is to ensure that the Government in power (the executive) cannot bribe an MP or MLA with more pay or executive powers to influence them.
Since a Prime Minister has to be a Member of Parliament, any potential PM candidate holding an "office of profit" is automatically disqualified from being an MP (and thus also the PM post).
But, there is currently no bar on an individual or industrialist becoming an MP or MLA even if they have lucrative government contracts. In fact, this has been a cause of concern highlighted by many politicians and the media as industrialists entering the Indian Parliament have been increasingly lobbying to get into Parliamentary Committees that frame industrial policies, and have even managed to do so to influence it. A most glaring example of this is the bidi baron and BJP MP, Shyama Charan Gupta:
... a member of the parliamentary committee on health claimed that there were no health risks associated with tobacco use. The mainstream media laughed at this suggestion. However, his comments were consequential; he was Shyama Charan Gupta, a BJP MP, known as the “bidi baron,” with an annual turnover of Rs. 200–250 crore in the bidi business. The health ministry heeded the recommendation of the parliamentary health committee not to display a warning covering 85 percent of tobacco products.
(In fact, even industrialist Anil Ambani is a former Member of Parliament).
Suggestions have actually been made to Parliament that industrialists with vested interests should not be allowed in Parliament Committees, but they are yet to be implemented:
In 2008, the Deo Committee set up to inquire into parliamentary misconduct said: “The committee members are of the view that a provision may be made to the effect that if a member has a personal, pecuniary or direct interest on any subject/matter, he should not be nominated in the first place to the Departmentally Related Standing Committee, which normally examines such subjects/matters” (Second Report of the Deo Committee, 2008). Some of the recommendations of the committee were accepted by the Rajya Sabha but have been ignored by the Lok Sabha.
(This is obviously very similar to the "office of profit" clause but only applicable to Parliamentary Committees, and does not seek to debar them from Parliament itself).
Article 102(1)(a) of Indian Constitution
Office of Profit In India
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