The ability to fine-tune band gap and band inversion in topological materials is highly desirable for the development of novel functional devices. Here we propose that the electronic properties of free-standing nanomembranes of the topological crystalline insulators (TCI) SnTe and Pb_1-xSn_x(Se, Te) are highly tunable by engineering elastic strain and membrane thickness, resulting in tunable band gap and giant piezoconductivity. Membrane thickness governs the hybridization of topological electronic states on opposite surfaces, while elastic strain can further modulate the hybridization strength by controlling the penetration length of surface states. We propose a frequency-resolved infrared photodetector using force-concentration induced inhomogeneous elastic strain in TCI nanomembranes with spatially varying width. The predicted tunable band gap accompanied by strong spin-textured electronic states will open new avenues for fabricating piezoresistive devices, infrared detectors and energy-efficient electronic and spintronic devices based on TCI nanomembrane.