Yes, oscillation can potentially damage your signal booster and even disrupt the cellular network[4]. The FCC requires all cell signal boosters to automatically detect oscillation and immediately reduce power or shut down to correct the condition[2][4]. If oscillation is not corrected, it can damage the cellular network[2].
Oscillation happens when the outside and inside antennas are too close, causing a feedback loop[1][4]. When a booster detects oscillation, it automatically reduces its gain, or signal boosting power, to correct the condition, which reduces the indoor signal coverage area[2]. If the reduced gain doesn't fix the oscillation, the booster must shut itself off[2]. Some boosters have indicator lights that will notify you when oscillation is occurring[6]. A solid red light on the amplifier indicates that oscillation has occurred and the signal booster has reduced power or shut down[4].
Citations:[1] https://www.youtube.com/watch?v=BiV4VtPzIi4
[2] https://www.weboost.com/blog/antenna-separation-and-why-its-important
[3] https://www.youtube.com/watch?v=4TNxmDNCJYw
[4] https://www.wilsonpro.com/blog/commercial-cell-phone-signal-booster-oscillation-and-overload
[5] https://www.pta.gov.pk/assets/media/ad_cons_paper_10102020.pdf
[6] https://powerfulsignal.com/support/knowledgebase/consideration/what-is-cell-signal-booster-oscillation/
[7] https://en.wikipedia.org/wiki/Parasitic_oscillation
[8] https://toshiba.semicon-storage.com/info/docget.jsp?did=59456
[9] https://escapees.com/rv-cell-signal-booster-review-and-installation-tips-2/