Context. With its 12 optical filters, the Javalambre-Photometric Local Universe Survey (J-PLUS) provides an unprecedented multicolor view of the local Universe. The third data release (DR3) covers 3192 deg2 and contains 47.4 million objects. However, the classification algorithms currently implemented in the J-PLUS pipeline are deterministic and based solely on the morphology of the sources. Aims. Our goal is to classify the sources identified in the J-PLUS DR3 images as stars, quasi-stellar objects (QSOs), or galaxies. For this task, we present BANNJOS, a machine learning pipeline that utilizes Bayesian neural networks to provide the full probability distribution function (PDF) of the classification. Methods. BANNJOS has been trained on photometric, astrometric, and morphological data from J-PLUS DR3, Gaia DR3, and CatWISE2020, using over 1.2 million objects with spectroscopic classification from SDSS DR18, LAMOST DR9, the DESI Early Data Release, and Gaia DR3. Results were validated on a test set of about 1.4 × 105 objects and cross-checked against theoretical model predictions. Results. BANNJOS outperforms all previous classifiers in terms of accuracy, precision, and completeness across the entire magnitude range. It delivers over 95% accuracy for objects brighter than r = 21.5 mag and ~ 90% accuracy for those up to r = 22 mag, where J-PLUS completeness is ≲ 25%. BANNJOS is also the first object classifier to provide the full PDF of the classification, enabling precise object selection for high purity or completeness, and for identifying objects with complex features, such as active galactic nuclei with resolved host galaxies. Conclusions. BANNJOS effectively classified J-PLUS sources into around 20 million galaxies, one million QSOs, and 26 million stars, with full PDFs for each, which allow for later refinement of the sample. The upcoming J-PAS survey, with its 56 color bands, will further enhance BANNJOS's ability to detail the nature of each source.