We present results on the emission-line properties of z = 1.4–7.5 star-forming galaxies in the Assembly of Ultradeep Rest-optical Observations Revealing Astrophysics (AURORA) Cycle 1 JWST/NIRSpec program. Based on its depth, continuous wavelength coverage from 1 to 5 μm, and medium spectral resolution (R ∼ 1000), AURORA includes detections of a large suite of nebular emission lines spanning a broad range in rest-frame wavelength. We investigate the locations of AURORA galaxies in multiple different emission-line diagrams, including traditional BPT diagrams of [O iii]λ5007/Hβ versus [N ii]λ6583/Hα, [S ii]λλ6717, 6731/Hα, and [O i]λ6300/Hα, and the ionization–metallicity diagram of [O iii]λ5007/[O ii]λ3727 (O₃₂) versus ([O iii]λ5007+[O ii]λ3727)/Hβ (R₂₃). We also consider a bluer rest-frame ionization–metallicity diagram introduced recently to characterize z > 10 galaxies, [Ne iii]λ3869/[O ii]λ3727 versus ([Ne iii]λ3869+[O ii]λ3727)/Hδ, as well as longer-wavelength diagnostic diagrams extending into the rest-frame near-IR: [O iii]λ5007/Hβ versus [S iii]λλ9069, 9532/[S ii]λλ6717, 6731 (S₃₂), and He iλ1.083 μm/Paγ and [S iii]λ9532/Paγ versus [Fe ii]λ1.257μm/Paβ. With a significant boost in signal-to-noise ratio and large, representative samples of individual galaxy detections, the AURORA emission-line diagrams presented here definitively confirm a physical picture in which chemically young, α-enhanced, massive stars photoionize the interstellar medium (ISM) in distant galaxies with a harder ionizing spectrum at fixed nebular metallicity than in their z ∼ 0 counterparts. We also uncover previously unseen evolution prior to z ∼ 2 in the [O iii]λ5007/Hβ versus [N ii]λ6583/Hα diagram, which motivates deep NIRSpec observations at even higher redshift. Finally, we present the first statistical sample of rest-frame near-IR emission-line diagnostics in star-forming galaxies at high redshift. In order to truly interpret rest-frame near-IR line ratios including [Fe ii]λ1.257 μm, we must obtain better constraints on dust depletion in the high-redshift ISM.