It probably depends on the state. E.g. Ohio code has:
§ 606.19 DERELICTION OF DUTY.
(A) No law enforcement officer shall negligently do any of the following:
(1) Fail to serve a lawful warrant without delay.
(2) Fail to prevent or halt the commission of an offense or to apprehend an offender, when it is in the law enforcement officer’s power to do so alone or with available assistance. [...]
(F) Whoever violates this section is guilty of dereliction of duty, a misdemeanor of the second degree.
I suspect that charges under laws like this are rather uncommon though (perhaps because of the need to prove negligence). It's much more common for the policeperson to be (just) fired.
But during the Floyd protests in Ohio, some policemen were charged with dereliction of duty. That also happened in an unrelated case in Cleveland. I haven't followed the cases tough, to see if there were convictions. From a quick look, these weren't simple charges of dereliction of duty, but the dereliction charge was added "on top" of different ones, like e.g. when the policeperson falsified a report.
Almost all such cases on a quick google search are from Ohio (I did find one in Florida, but it involved the policeman refusing to answer a call--and no charges were filed), so perhaps similarly precise laws don't exist in other US states. There are broad/vague statues on dereliction of duty of officials in some other states, but seemingly not as precise on law enforcement as in Ohio.
In other cases like West Virginia's Code §8-14-3, removal from office (for "official misconduct") is the only recourse specified in the law:
The mayor and police officers of every municipality and any municipal sergeant shall aid in the enforcement of the criminal laws of the state within the municipality, independently of any charter provision or any ordinance or lack of an ordinance with respect thereto, and to cause the arrest of, or arrest, any offender and take him or her before a magistrate to be dealt with according to the law. Failure on the part of any such official or officer to discharge any duty imposed by the provisions of this section is official misconduct for which he or she may be removed from office.
In general, conducting cross-state law comparison/research in the US is a nightmare...
And comments about other common law jurisdiction in other answer aren't entirely correct. Since 2015 the UK has an explicit law that punishes "improper exercise of police powers", which is pretty broadly construed, including failure to exercise the powers (to the benefit of a 3d party) when the "a reasonable person" would have expected otherwise. I'm not aware of any prosecutions under this law for failure to exercise though. (The more general common law offence of "misconduct in public office" also applies to 'nonfeasance' acts.
Actually, the common law offence was invoked in a case of inaction by police, circa 2013-2016.)
Somewhat related, "failure to intervene" is fairly limited in US federal-level law/jurisprudence, basically limited to failing to prevent a civil rights offense by another police [or similar] officer:
The framework for establishing a successful Section 1983 claim of failure to intervene may vary slightly by jurisdiction, however the premise is substantially the same. In the Seventh Circuit, “an officer who is present and fails to intervene to prevent other law enforcement officers from infringing the constitutional rights of citizens is liable under § 1983 if that officer had reason to know: (1) that excessive force was being used, (2) that a citizen has been unjustifiably arrested, or (3) that any constitutional violation has been committed by a law enforcement official; and the officer had a realistic opportunity to intervene to prevent the harm from occurring.” Yang v. Hardin, 37 F.3d 282, 285 (7th Cir. 1994).
(Worth recalling here perhaps that that section 1983 is a civil recourse to begin with, not grounds for criminal charges.)
Additionally there are some cases where police was successfully sued under section 1983 for failing to enforce restraining orders related to domestic violence, the first of these was Thurman v. City of Torrington. Subsequently a number of
states have passed laws that have made police arrests mandatory in domestic violence and restraining order cases. At least 23 US states now have these. But for the purpose of your question... I have no idea what the punishement for the officer failing to follow such laws would be. I guess just dismissal from job in most (in not all) states, but that's just a guess.
In some states like Washington (which has such a mandatory-arrest domestic violence law -- RCW 10.31.100(2)(c)), failure of a public official to do their duties is automatically a misdemeanor (under RCW 42.20.100), but I'm not aware of cases where this provision was used when police didn't implement mandatory arrests.