What's the best way to burn a new firmware image (.hex file) on a STM32 target in the field? We don't need to debug. I need to be able update the device by sending a .hex file to the tech and he needs to burn it.
Depends on mode of connectivity. For USB you can use a .DFU file and DFU tools. For Serial the Flash Loader Demonstrator. For SWD/JTAG there is ST-LINK Utilities.
More ideally for field updates you own your update process, where you have a boot loader using the standard interfaces you provide the customer, and up load via you own application or X/Y-Modem in via a terminal. See IAP examples.
There is also this new tool
STM32CubeProgrammer - New All-in-one Software Tool
Debuggers like stlink or jlink can do what you want.
Or Google production programmer.
the tech ... needs to burn it
Be more specific about "the tech"
Could she be equipped with an ST-Link and laptop, and just use that?
A genuine ST-Link is only twenty quid or so; cheap (but perfectly usable) knock-offs are available on ebay and the like for under a fiver:
Retrieving data ...