Not sure if this helps, but the only place I can find anything about DME being required is when operating above 24k if you dont have a suitable RNAV.
Below 24, you need navigation equipment for the route to be flown, so if flying off airways via VOR radial you need a VOR, but I dont see anything about DME.
Hope that helps.
91.205
(d)
Instrument flight rules. For IFR flight, the following instruments and equipment are required:
(1) Instruments and equipment specified in paragraph (b) of this section, and, for night flight, instruments and equipment specified in paragraph (c) of this section.
(2) Two-way radio communication and navigation equipment suitable for the route to be flown.
(3) Gyroscopic rate-of-turn indicator, except on the following aircraft:
(i) Airplanes with a third attitude instrument system usable through flight attitudes of 360 degrees of pitch and roll and installed in accordance with the instrument requirements prescribed in §121.305(j) of this chapter; and
(ii) Rotorcraft with a third attitude instrument system usable through flight attitudes of ±80 degrees of pitch and ±120 degrees of roll and installed in accordance with §29.1303(g) of this chapter.
(4) Slip-skid indicator.
(5) Sensitive altimeter adjustable for barometric pressure.
(6) A clock displaying hours, minutes, and seconds with a sweep-second pointer or digital presentation.
(7) Generator or alternator of adequate capacity.
(8) Gyroscopic pitch and bank indicator (artificial horizon).
(9) Gyroscopic direction indicator (directional gyro or equivalent).
(e)
Flight at and above 24,000 feet MSL (FL 240). If VOR navigation equipment is required under paragraph (d)(2) of this section, no person may operate a U.S.-registered civil aircraft within the 50 states and the District of Columbia at or above FL 240 unless that aircraft is equipped with approved DME or a suitable RNAV system. When the DME or RNAV system required by this paragraph fails at and above FL 240, the pilot in command of the aircraft must notify ATC immediately, and then may continue operations at and above FL 240 to the next airport of intended landing where repairs or replacement of the equipment can be made.