Would an adhesive surface have more air resistance?

Imagine spreading double-sticky tape all over the surface of a car or a plane. Would there more significantly more aerodynamic drag as a result of the adhesive 'sticking' to air molecules and slowing down? This would certainly cause more resistance in solid mediums but would this cause more resistance in air or water? If an adhesive surface can drastically increase friction with solid surfaces, could one also increase friction with the air?

It seems the question assumes that because the tape is sticky, it will somehow strike or capture air molecules which will increase drag. This is incorrect since air molecules have very weak intermolecular forces (you might get a layer of gas molecules to form on the tape) and are moving too quickly to begin with to stick. Also, even though it's a sticky surface, it's still fairly flat/smooth.

Aerodynamic drag does depend on the roughness of a surface$$^1$$, and even though objects with a rough surface will have more drag, it's hard to imagine that a sticky taped surface that is more or less smooth (though sticky), has an appreciable skin friction coefficient (see below). And as pointed out by Niels, objects moving through a gas have a boundary layer adjacent to the surface which is more or less stationary relative to the surface. For surfaces with skin roughness (see links below) enough to noticeably affect the boundary layer profile, you would expect the surface to be a lot more rough than sticky tape. So sticky tape would not have the effect you ask of, in air.

You're right that the same cannot said about solids, but solids that have sticky tape will experience cohesive/adhesive forces when in contact.

$$^1$$ This is referred to as skin friction where in the equation for drag force, skin friction is included in the drag coefficient $$c_D$$ in the first above linked equation, where the total skin friction drag force can be computed by $$F=\iint \limits _{S}c_{f}{\frac {\rho v^{2}}{2}}dA$$ where $$c_f$$ is the skin friction coefficient.

for the case of a sticky body moving through air:

Making the surface of the object sticky will have no influence on the thickness of the boundary layer or on the drag force exerted upon the moving body by the air that surrounds it.

This is because there always exists a boundary layer of air right next to the moving object which is already "stuck" to its surface, at zero velocity. As you move away from the object, the airstream velocity increases from zero to its full value some distance away from the object.

Both answers sensibly focus on the low-Knudsen number flow regime typically experienced by the cars and planes mentioned in the question. If your interest extends to vehicles like low-altitude satellites and ICBMs you will have to consider another regime that occurs when the mean free path of the molecules in the fluid is >10x the characteristic length of the object moving through that fluid. In "free molecular flow" there is no boundary layer; molecules of the incoming gas ballistically impact the object and are either elastically reflected, or re-emitted after gas-surface interactions (GSI). The drag on a body due to the fraction of particles that are adsorbed and re-emitted does depend on the 'stickyness' of the surface, but how so is not straightforward and depends on a wide range of parameters:

The scattering behaviours observed are not constant as they tend to change substantially with the system considered and, in particular, with the ratio between the mass of the gas particles and the surface atoms, the range of interaction, the energy/temperature of the incident beam with regards to the surface temperature, the molecular or atomic species involved in the experiment, the presence of adsorbents on the target surface, the morphology of the surface considered despite the level of roughness and the relative position and orientation of the gas particles and surface atoms [97].

Experiments investigating these phenomena typically fire a beam of gas at a metallic target and measure the location and energy of the scattered beam.

I'm posting this answer mainly as a curiosity, but there is extensive literature if you're interested. Here's the review the above quote is from.