NY state here.
If I go to get a second opinion from a physician other than my primary, are there any regulations (HIPAA or otherwise) that require that 2nd physician to share consultation notes/lab results with my primary?
Does the HIPAA Privacy Rule permit doctors, nurses, and other health care providers to share patient health information for treatment purposes without the patient’s authorization?
Yes. The Privacy Rule allows those doctors, nurses, hospitals, laboratory technicians, and other health care providers that are covered entities to use or disclose protected health information, such as X-rays, laboratory and pathology reports, diagnoses, and other medical information for treatment purposes without the patient’s authorization. This includes sharing the information to consult with other providers, including providers who are not covered entities, to treat a different patient, or to refer the patient. See 45 CFR 164.506.
This implies that if Doctor 1 requests your records from Doctor 2 for treatment purposes, Doctor 2 may provide them. Reciprocal practices in the medical field may make such requests required in practice, if anyone at Doctor 2's practice might in the future want to request records/cooperation from any practice in Doctor 1's network.
This rule also implies that if Doctor 2 believes that sharing the information with Doctor 1 will be helpful for treatment purposes, Doctor 2 can do that without a request from Doctor 1, and depending on the situation may be ethically required to do so if Doctor 2 believes it's critical for Doctor 1 to get this information.
If you are requesting that Doctor 2 share the information with Doctor 1, and Doctor 2 is refusing, you can get a copy of your record and give it to Doctor 1. User Nate Eldredge has pointed out in his subsequent answer that 45 CFR 164.524 (c)(3)(ii) permits you to skip a step and direct Doctor 2 to transmit the health records directly to Doctor 1 instead of going through you directly, but from a legal perspective it's basically the same. Doctor 2 can charge a fee for this.
Absent any obvious reason why Doctor 2 would feel ethically obliged to inform Doctor 1 about without treating/taking care of that reason directly, and absent any request for the information to be shared (the question as stated at time of answering does not indicate that any request has been made by anybody for the information to be shared), there is no law requiring Doctor 2 to send the report to anybody.
This answer assumes that Doctor 1 and Doctor 2 work for different healthcare systems, such that Doctor's 2 entry of your record which may automatically go to his/her employer (by contract law) does not therefore also reach the same system Doctor 1 accesses when looking up your record, automatically.