The argument has seemingly bifurcated here. The original discussion pertained to religious organizations and if by endorsing a certain candidate or position they are violating the First Amendment, thereby giving cause to have their tax-exempt status revoked. Now we seem to be discussing if tax-exempt organizations in general, like Planned Parenthood and MoveOn.Org, should lose their exemption status for participating in an election process.
According to the Tax Code Care posted, they should.
No there are separate laws that cover non profits like Moveon.org. The claim that just being tax exempt prevents political participation is false.