I am wiring a Tesla wall connector with #6 gauge THHN in conduit and installing a 60 amp breaker to allow for a max 48 amp charge. If I only need a 48 amp charge every so often and choose to continually set the car charger to charge using a lower amount of amps (such as 16 or 24 or 32 amps) will that create any issues being on a 60 amp breaker?
First, in order to provision 60A to your circuit, you need to have 60A of headroom in your NEC 220.82 Load Calculation. (there's more than one way to do a Load Calc in Article 220, but 220.82 is both simpler and as a rule more favorable.) If you don't, then choose a smaller circuit, or install an EVEMS such as Wallbox or Emporia.
The wire must be at least large enough for the breaker; per Table 310.15(B)(16) - and you use 75°C column for THHN and most wires. 60°C for NM/UF aka "Romex"*.
The Tesla Wall Connector has a "hard / firm" setting which is set at installation. That must be <= the wire capacity *. This setting is intentionally difficult to change per UL/CSA requirements for safety. Further, the unit nameplate must be physically labeled with this new setting. ***
So, to answer your question -- Once that "firm" setting is installed, the EVSE is welcome to provide a "user-facing" interface to allow the end user to reduce charge current further (but not increase it above the firm setting, obviously).
An example of when you might do that is the known-faulty charge ports on the Hyundai group cars, where common wisdom is don't push them past 32A or they melt, then you must use your remaining charge to limp to a dealer for a (in-warranty but annoying) port replacement.
In your case, running 65A wire to a "60A / 48A actual from the factory" Tesla Wall Connector requires no modification of the default firm settings, and you are welcome to use either the car's controls or whatever "app" comes with an EVSE to lower the charge rate further.
* For readers passing by, there's a huge asterisk for NM or UF cable (aka Romex). Because of the cheapness of the insulation (oddly not reflected in the price of the cable), it uses the 60°C thermal rating. And here, 6 AWG is an odd duck, because it is only 55A - not enough for a nominal 60A breaker / 48A actual charge rate. #6 NM/UF requires setting the EVSE for 50A breaker / 40A charge rate**. This often confuses people because 55A is not a "standard" breaker size, so NEC allows you to "round up" to the next available breaker size, 60A - But, you must treat it in all respects like it is a 55A breaker. It is NOT a 60A breaker for purposes of configuring an EVSE!
** 55A breaker for 44A charge rate would be acceptable, but no EVSE known offers that as an option. This is one of many stupidities in EV home charging design - notice in Table 310.15(B)(16), there is no such thing as 60A wire. The 230V world doesn't have 60A either, their size is 63 or 64 amps. So it makes zero sense for 60A to be the de-facto standard for level 2 charging.
*** The electrician relies on this alteration of the nameplate label to determine the size of circuit to run, and the inspector relies on it to determine whether the circuit was run correctly. E.G. an unmodified Tesla Wall Connector on 6/2 Romex is a code violation since the unit is nameplated 60A but on a 55A circuit.
All a breaker cares about is that the current never exceeds the rating. You will find that when your car is not plugged in the Tesla charger will be pulling a few milliamps to run the local electronics trying to detect the car and provide whatever smarts it does.
Your breaker will have no issue supplying anything between 0 and the rated current.