When Rush sings that "his mind is not for rent / To any God or government", they compare the mind to a place you can live in or use in exchange for some form of payment. What the "payment" would be is left unstated; in the case of religion, it is typically some form of heaven or paradise you might go to after death; in the case of government, it may be government functions such as social security, health care, education etc. But, as I said, this is left unstated.
When you rent a place, you can typically redecorate it. Rush's Tom Sawyer does not want a god or a government to "redecorate" his mind; he does not want to relinquish control over his mind to a god or government.
Does this mean the song "promotes" an "anti-religious view"? That depends on how you defined these words.
Wiktionary defines promote as "To advocate or urge on behalf of (something or someone); to attempt to popularize or sell by means of advertising or publicity" and "To encourage, urge or incite". To promote something clearly goes beyond merely expressing it. The song expresses resistance to control by a god or a government, but it does not say that others should take the same stance. However, it can still be maintained that Rush promotes these ideas by pointing out how songs can work in society and that successfully and convincingly expressing certain ideas essentially comes down to promoting them.
Finally, there is the question whether those lines are anti-religious. Wiktionary defines antireligious as "Opposed to religion". This is both vague and ambiguous. One can be opposed to religion in one's personal life (i.e. rejecting it personally without rejecting it in society) or against religion in general (i.e. religion should not exist it all). Having said this, "antireligious" is still ambiguous in a different way, because "religion" is ambiguous. According to Wiktionary, religion can mean both "The belief in and worship of a supernatural controlling power, especially a personal god or gods" and "Any practice to which someone or some group is seriously devoted". (Wiktionary also lists a few other meanings, but the two meanings listed here should be sufficient to illustrate a specific ambiguity.) Some people, e.g. Peter Boghossian, have argued that these concepts should be clearly distinguished; the first concept should be called faith, while the community aspect should be called "religion". From this point of view, the above lines from Rush's "Tom Sawyer" are anti-faith (on a personal level, at least) but not anti-religious.