The aim of this article is to provide an analysis of the influence of the type of hard anti-wear coatings on the friction behaviour of DC01 deep-drawing steel sheets. DC01 steel sheets exhibit high formability, and they are widely used in sheet metal forming operations. The tribological properties of the tool surface, especially the coating used, determine the friction conditions in sheet metal forming. In order to carry out the research, this study developed and manufactured a special bending-under-tension (BUT) friction tribometer that models the friction phenomenon on the rounded edges of tools in the deep-drawing process. The rationale for building the tribotester was that there are no commercial tribotesters available that can be used to model the phenomenon of friction on the rounded edges of tools in sheet forming processes. The influence of the type of coating and sheet deformation on the coefficient of friction (CoF) and the change in the topography of the sheet surface were analysed. Countersamples with surfaces prepared using titanium + nitrogen ion implantation, nitrogen ion implantation and electron beam remelting were tested. The tests were carried out in conditions of dry friction and lubrication with oils with different kinematic viscosities. Under dry friction conditions, a clear increase in the CoF value, with the elongation of the samples for all analysed types of countersamples, was observed. Under lubricated conditions, the uncoated countersample showed the most favourable friction conditions. Furthermore, oil with a lower viscosity provided more favourable conditions for reducing the coefficient of friction. Within the entire range of sample elongation, the most favourable conditions for reducing the CoF were provided by uncoated samples and lubrication with S100+ oil. During the friction process, the average roughness decreased as a result of flattening the phenomenon. Under dry friction conditions, the value of the Sa parameter during the BUT test decreased by 20.3–30.2%, depending on the type of countersample. As a result of the friction process, the kurtosis and skewness increased and decreased, respectively, compared to as-received sheet metal.