If there no overlap, then invertibility is lost, and so is information.
If there is overlap, but it is little, then analysis information is lost (but not synthesis, which is more fundamental). Namely, from 0 to 0.5 times the sampling frequency, if "hop length" is 2, then analysis information for frequencies bewteen 0.25 and 0.5 is aliased. If hop length is 4, then 0.125 to 0.5 is aliased, and so on.
"Analysis information" is the very goal of transforming data in the first place (if all we cared for was preserving the input, simply don't do a transform). For STFT, it refers to time-frequency modulation information: a row captures (within the window's resolution) the instantaneous frequency and amplitude of the signal over time. The lesser the overlap (greater hop length), the less dynamics are captured.
The good news is, if all you seek is the spectrogram, i.e. absolute value of STFT (amplitude), then loss of this information is fairly minimal: taking absolute value globally shifts frequencies of every row toward low frequencies, and most often very low frequencies, which permits a generous hop length. This is proven for the wavelet transform, but the arguments will approximately hold for most STFT windowings.
Worth noting, strict invertibility is likewise lost for the spectrogram for any hop length other than 1; the extent of loss can be estimated via inversion algorithms like Griffin-Lim. However, if we cannot invert the raw STFT, it literally means we completely threw away segments of data; this is 100% irrecoverable unlike the modulus.
If processing power is a limitation, options are: 1) use a wider window, which increases frequency resolution at expense of time resolution but also permits greater hop size; 2) overlap-add.