Not unless it’s very severe.
They can’t detect when a Time-Turner has merely been used, or there’d be much more suspicion on Hermione after Sirius had escaped. So I think they can’t detect minor infractions. With no way to audit every use of the Time-Turner, it has to be a trust-based system. Or rather, trust augmented with the fear of the consequences.
Major damage is easier to spot. For example, Pottermore tells the tale of Madam Mintumble, who travelled back from 1899 to spend five days in 1402. Her tampering was so severe, 25 people were “un-born”, and time was severely disrupted in the present:
Finally, there were alarming signs, during the days following Madam Mintumble’s recovery, that time itself had been disturbed by such a serious breach of its laws. Tuesday following her reappearance lasted two and a half full days, whereas Thursday shot by in the space of four hours. The Ministry of Magic had a great deal of trouble in covering this up and since that time, the most stringent laws and penalties have been placed around those studying time travel.
Madam Mintumble aged five centuries in the process and died shortly after returning to the present. I expect that would scare anybody feeling more reckless.
And any major breach of the law would probably cause similarly noticeable effects on time in the “present”. Your list of suspects is fairly small, and that would be both detectable and punishable. It sounds like Time-Turners are fairly difficult to manufacture, so the Ministry effectively monopolises the stock (at least in Britain), hence a short suspect list.
