I get in to NYC on a regular basis.
Tunnel clearance in the Lincoln Tunnel is 13'. Holland Tunnel is 12'6" I don't get to the others much at all. jplante4 is correct about LP tanks - forbidden in both of the above tunnels.
Best time to go IMHO if you just want to motor down 5th Ave is Saturday or Sunday morning PRIOR to 10 a.m. Regardless of when you go, plan on taking your time. There is always construction going on in the city and it is not unusual for streets to be restricted down to one lane (7th Ave. between about 42nd and 45th being one of them). Check your mirrors always.
Couple of notes: NO right turn on red in the city - EVER. Pedestrians assume you will stop for them, even if they are crossing against the light. This is because 1) the traffic lights hold red in all directions for pedestrian traffic before any green, and 2) many of them are buried in their smartphones with earbuds in. Beware motorcycles and bicycles - they will ride between you and the next lane even if the space is no wider than their handlebars. You so much as brush them and it will be deemed your fault. Same with pedal-powered rickshaws (they are the worst!). Check your mirrors always.
Parking - if you decide you want to get out and walk, well, good luck. Most parking garages will not have enough clearance for you. Tempting as it may be do not park in zones marked for MTA (NYC transit), non-MTA, or loading/unloading. Parking fines in the city START at $115 and they are diligent about enforcement (it's a revenue stream for the city).
IMHO outside of Times Square the famous avenues are not all they're cracked up to be. A nice ride - and it will be quick - is 12th Ave./West Side Hwy along the Hudson past the cruise ship terminals. If you are too close to the famous buildings you can't see much from your cab anyway, because you'll be too busy checking your mirrors.
Better yet - Park at Liberty State Park in NJ and take the ferry to Ellis Island, Statue of Liberty then over to Manhattan. The Park is easy to get to right off of I-78. The ferry to Manhattan takes you to the lower end where a lot of the attractions are. Good luck and safe travels. Did I mention to check your mirrors always?