Rules as written: Yes - only spell slots are mentioned
The description of the simulacrum spell simply states:
The simulacrum lacks the ability to learn or become more powerful, so it never increases its level or other abilities, nor can it regain expended spell slots.
Besides HP, the only expendable resource specifically mentioned in the spell is spell slots. Thus, by a reading of the rules as written alone, one could argue that other class features (such as a sorcerer's sorcery points, a fighter's uses of Action Surge, a barbarian's uses of Rage, and so on) are technically not prevented from being refreshed on a long rest.
Rules as intended: No, a simulacrum's expended resources remain expended
Jeremy Crawford answered this exact question (and addressed the general case) in an unofficial tweet in June 2018:
3 Yes/No questions about Simulacrum after it does a long rest: 1) can Simulacrum-Sorcerer recover sorcery points and spend them to create spell slots? 2) can Simulacrum-Fighter get back an Action Surge use? 3) can a Simulacrum-Warlock get back Mystic Arcanum uses?
Simulacrum—the spell's creation can't gain levels or regain expended spell slots (RAW). The creation is meant to be unable to regain use of any of its features that it expends (RAI).
Well that RAI is a huge difference. It means making copies of battle masters, bards, monks, etc are way worse. I'm glad this isn't an errata.
There's a good chance it will be. A simulacrum is meant to lose efficacy over time, essentially running out of juice, until only at-will abilities remain.
Even though, as written, the inability to regain things only applies to spell slots, Crawford makes the intent clear here: Any of the simulacrum's expended resources remain expended, and can not be regained.
(Note: the simulacrum spell was actually changed in the 2018 PHB errata, but not to fix this loophole. The only change to the spell was to specify that the illusory duplicate is a construct, rather than a beast or humanoid.)