The hard part is usually breaking into the program - using perhaps what you refer to as a "programmer's reset" - so I'll add some notes about that below for posterity.
To save the HIRES screens you want to be able to either reset to a command prompt first, or directly reboot to DOS 3.3 with HIRES memory intact. Then you can save HIRES page 1 or 2 with:
BSAVE HIRES1,A$2000,L$1FF8 or
To load them back and view fullscreen:
Here is a summary of the main ways to get to a command prompt with HIRES intact:
On a II or II Plus, use the Old Monitor ROM (maybe in a Firmware
Card or Language Card)
On a II/II Plus or //e use a Senior Prom or Wildcard (if the software doesn't check for a Wildcard)
On a IIgs use the Visit Monitor CDA
Boot code trace the disk until the HIRES image is in memory, or hack the disk directly by searching for and modifying the code which sets the soft reset vector (not covered here)
Image the disk and use an emulator (the easiest option today - again, not covered here)
Reset Details & Miscellania
Reset on the Apple II, or Ctrl-Reset on the II Plus, //c and //e, causes the 6502 CPU to jump to the address stored in the reset vector at $FFFC/$FFFD.
In the original (Old Monitor) Apple II ROM the reset vector points to a reset handler that falls through to the monitor entry point, so reset will enter the monitor and the HIRES pages will be intact for saving.
In an Apple II or II Plus with an Autostart ROM the reset vector points to a reset handler which checks the "soft" reset vector at $3F2-$3F4. If $3F4 == $3F3 EOR #$A5 then it jumps to the address in $3F2/$3F3, otherwise it reboots. Software almost always used the soft reset vector to either prevent exit, or to clear memory and reboot.
If a Firmware Card is installed then you can use the switch on the back of the card to select the Integer Basic/Old Monitor ROM so reset will enter the monitor.
If a 16k RAM Language Card is installed, reset doesn't affect its state, so if high RAM ($D000-FFFF) is switched in when a reset occurs the reset vector is fetched from RAM. Software of that era often took advantage of this, switching in RAM and overwriting the reset vector, making it impossible to reboot the machine without turning of the power.
If the software doesn't overwrite the reset vector, you can use that trick yourself. From the monitor:
C081 C081 // write enable high RAM
F800<F800.FFFFM // copy the monitor from ROM to RAM
FFFC:59 FF // point the reset vector to the Old Monitor ROM reset handler
C080 // read enable RAM
Now boot the software and press Reset to enter the monitor, then boot DOS and save HIRES.
In the //e and //c Apple added the Ctrl-OpenApple-Reset feature (referred to as "JIVE" in the source code) to ensure that the user could always reboot the system. While this is exactly the behaviour we would want, it also runs a routine called "BLAST" which writes two bytes to each memory page from BF down to 02, making this feature useless to pirates cracking software - and to us for saving HIRES.
Reset in these machines also resets the emulated 16k RAM Language Card state to read main ROM so software can't hijack the hard reset vector and neither can we - so we can't use the trick above.
If the HIRES page you want to capture is mostly black, you could use Ctrl-OpenApple-Reset and then manually fix up HIRES. The following routine will zero out the bytes that BLAST wrote to HIRES1:
BL EQU $3C
BH EQU $3D
BLOOP STA BH
Other hacking options include boot-code tracing the software to get a HIRES title screen, or searching the disk for code that sets the soft reset vector and changing it to reset into the monitor.
The only remaining options are to use additional hardware. You can build or order a Senior Prom (or other hacking ROMs) for any machine, use a Wildcard (or other copy card), or use an emulator.