Optical sensing and imaging methods for biomedical applications, such as spectroscopy and laser-scanning fluorescence microscopy, are incapable of performing sensitive detection at high scan rates due to the fundamental trade-off between sensitivity and speed. This is because fewer photons are detected during short integration times and hence the signal falls below the detector noise. Optical postamplification can, however, overcome this challenge by amplifying the collected optical signal after collection and before photodetection. Here we present a theoretical analysis of the sensitivity of high-speed biomedical sensing and imaging systems enhanced by optical postamplifiers. As a case study, we focus on Raman amplifiers because they produce gain at any wavelength within the gain medium's transparency window and are hence suitable for biomedical applications. Our analytical model shows that when limited by detector noise, such optically postamplified systems can achieve a sensitivity improvement of up to 20 dB in the visible to near-infrared spectral range without sacrificing speed. This analysis is expected to be valuable for design of fast real-time biomedical sensing and imaging systems.