Pulling membrane nanotubes from liposomes presents a powerful method to gain access to membrane mechanics. Here we extend classical optical tweezers studies to infer membrane nanotube dynamics with high spatial and temporal resolution. We first validate our force measurement setup by accurately measuring the bending modulus of EPC membrane in tube pulling experiments. Then we record the position signal of a trapped bead when it is connected, or not, to a tube. We derive the fluctuation spectrum of these signals and find that the presence of a membrane nanotube induces higher fluctuations, especially at low frequencies (10-1000 Hz). We analyse these spectra by taking into account the peristaltic modes of nanotube fluctuations. This analysis provides a new experimental framework for a quantitative study of the fluctuations of nanotubular membrane structures that are present in living cells, and now classically used for in vitro biomimetic approaches.