This study aimed to find relationships between the properties of beeswax-based oleogels and the type of oil used. The influence of linseed, sunflower, olive, and fish oils was studied. For these oils, the fatty acid composition, the content of total polar components, and the iodine value were characterized. Textural and thermodynamic properties were determined for oleogels, the oil-binding capacity was estimated, and the morphology of crystals was studied. The concentration of beeswax in all oleogels was 6.0% w/w. It was shown that the type of oil has a significant influence on all characteristics of the oleogels. The use of different oils at the same technological treatment leads to the formation of crystals of diverse morphology—from platelets to spherulites. At the same time, it was revealed that some characteristics of oils have a varying contribution to the properties of oleogels. The content of total polar materials in oils is associated with a decrease in strength parameters (yield value and elastic modulus) and the oil-binding capacity of oleogels. In its turn, the iodine value of oils has a close positive correlation with the melting and crystallization temperatures of oleogels. The results obtained in this article indicate that the properties of beeswax-based oleogels can be directed by changing the oil composition.