Normally the design basis for tube wall thickness of refinery process heaters coils, are based on API RP 530 for 100.000 hours service. Structural parts of the heater could be built according to ASME.
Although you do not specify which ASME , I doubt if structural parts are designed according any ASME . ASME Codes deal with pressure parts.
API 530/560 used for refinery heaters gives you the option to design for shorter as well as longer than 100.000 hrs as well as short term conditiond at higher temperatures.(E.g. during decoking).
ASME 1 and ASME VIII-1 only use 100.000 hrs with a max. allowable temperature.
Heaters can be supplied in accordance with most design codes such as ASME, API, DIN, BS, AS and many others.
Process piping is normally built according to ASME B31.3.
Boilers are normally built according to ASME.
Tube thickness for process heaters are normally in accordance with API-RP530 for 100000Hours.