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Scanning-tunneling spectroscopy experiments performed on conjugated polymer films are compared with three-dimensional numerical model calculations for charge injection and transport. It is found that if a sufficiently sharp tip is used, the field enhancement near the tip apex leads to a significant increase in the injected current, which can amount to more than an order of magnitude and can even change the polarity of the predominant charge carrier. We show that when charge injection from the tip into the organic material predominates, it is possible to probe the electronic properties of the interface between the organic material and a metallic electrode directly by means of tip height versus bias voltage measurements. Thus, one can determine the alignment of the molecular orbital energy levels at the buried interface, as well as the single-particle band gap of the organic material. By comparing the single-particle energy gap and the optical absorption threshold, it is possible to obtain an estimate of the exciton binding energy. In addition, our calculations show that by using a one-dimensional model, reasonable parameters can only be extracted from $z\text{\ensuremath{-}}V$ and $I\text{\ensuremath{-}}V$ curves if the tip apex radius is much larger than the tip height. In all other cases, the full three-dimensional problem needs to be considered.