We present tip-enhanced upconversion photoluminescence (PL) images of Er(3+)- and Yb(3+)-doped NaYF4 nanocrystals on glass substrates with subdiffraction spatial resolution. Tip-sample distance dependent measurements clearly demonstrate the near-field origin of the image contrast. Time-resolved PL measurements show that the tip increases the spontaneous emission rate of the two emission channels of Er(3+) in the visible region. Very efficient enhancement of upconversion PL is discussed in the context of the two-photon nature of the excitation process and homoenergy transfer between the ions within the nanocrystals. Comparison between different nanocrystals and tips shows a strong influence of the tip shape on the image contrast that becomes particularly relevant for the larger dimensions of the investigated nanocrystals.