Substituting noble metals for high-index dielectrics has recently been proposed as an alternative strategy in nanophotonics to design broadband optical resonators and circumvent the Ohmic losses of plasmonic materials. In this paper, we demonstrate that subwavelength silicon nanoantennas can manipulate the photon emission dynamics of fluorescent molecules. In practice, we show that dielectric nanoantennas can both increase and decrease the local density of optical states at room temperature, a process that is inaccessible with noble metals at the nanoscale. Using scanning probe microscopy, we analyze quantitatively, in three dimensions, the near-field interaction between a 100-nm fluorescent nanosphere and silicon nanoantennas with diameters ranging between 170 and 250 nm. Associated with numerical simulations, these measurements indicate increased or decreased total spontaneous decay rates by up to 15% and a gain in the collection efficiency of emitted photons by up to 85%. Our study demonstrates the potential of silicon-based nanoantennas for the low-loss manipulation of solid-state emitters at the nanoscale and at room temperature.