Can wing-tip vortices be reduced/eliminated with a rear-facing propeller near the wing-tip?
Yes. From an aerodynamic standpoint it makes some sense to use the wingtip vortex in combination with a propeller.

In this article by Sinnige et al., the researchers modeled and tested a propeller in various positions along a semispan, concluding that there were real gains to be had due to the increased span efficiency. Up to 15% less drag was measured when the propeller was positioned at the wingtip compared to a conventional mounting:


  By positioning the propeller at the tip of the wing, the slipstream interacts with the flow around the wingtip, thus affecting the roll-up and downstream behavior of the wingtip vortex. PIV measurements downstream of a propeller–wing model showed that this leads to a reduction in overall swirl with inboard-up rotation, for which the swirl in the slipstream is opposite to that associated with the wingtip vortex. At the same time, the system performance was found to improve due to a reduction of the wing induced drag, leading to the conclusion that the decrease in swirl causes a reduction in downwash experienced by the wing.
  
  Apart from the change in drag, the interaction of the wing with the propeller slipstream also modifies the wing lift. The locally enhanced dynamic pressure increases the lift over the spanwise part of the wing washed by the slipstream, which is amplified by the induced swirl for the case with inboard-up rotation. As a result, a strong spanwise variation in lift occurs with the propeller on. The induced velocities caused by this lift gradient lead to a spanwise shearing of the slipstream. With outboard-up rotation, the swirl in the slipstream acts to locally oppose the increase in wing lift due to the propeller-induced dynamic-pressure rise. Compared to the inboard-up rotation case, this leads to a reduction in wing lift at a given angle of attack, thus also a reduction in maximum lift coefficient. Furthermore, the direction of the spanwise shearing of the propeller slipstream is inverted on both sides of the wing.
  
  To quantify the potential aerodynamic benefits of the wingtip-mounted configuration, a direct comparison was made with a conventional configuration, with the propeller mounted on the inboard part of the wing. The increase in wing lift due to the interaction with the propeller was 1 - 4% smaller for the wingtip-mounted configuration than for the conventional configuration. For the latter, the enhanced dynamic pressure and swirl in the slipstream acts over double the spanwise extent, and on a part of the wing where the section lift is higher than for the wingtip-mounted configuration. At higher angles of attack, the lift advantage for the conventional configuration could be further enhanced
  by the local angle-of-attack increase in proximity of both sides of the nacelle.
  
  In terms of drag performance, on the other hand, the wingtip-mounted configuration showed superior performance. At a wing lift coefficient of $C_L = 0.5$ and a thrust coefficient of $0.09 &lt; C_T &lt; 0.13$, the drag reduction amounted to about 15 - 40 counts (5 - 15%) compared to the conventional configuration. The aerodynamic benefit of the
  wingtip-mounted configuration further increases with increasing wing lift coefficient and propeller thrust coefficient, leading to a drag reduction of 100 - 170 counts (25 - 50%) at $C_L = 0.7$ and $0.14 &lt; C_T &lt; 0.17$. An analysis of the wing performance confirmed that this drag benefit is mostly due to a reduction of the wing-induced drag. Compared to the conventional configuration, a relative increase in span efficiency of up to 40% was measured for the wingtip-mounted configuration. Although the exact drag benefit will be specific to vehicle design and operating conditions, it is concluded that the interaction between the propeller slipstream and the wingtip vortex leads to a clear drag reduction for the wingtip-mounted configuration.


Keep in mind this is purely aerodynamical, of course, and from a structural standpoint things might not be so clear-cut. While weight in the wings helps alleviate the bending stresses generated by lift, a large mass at the tip can lower the first bending eigenfrequency to unacceptable levels, risking a coupling of the wing oscillations with some other mode in flight or on the ground.

As far as I am aware wingtip fuel tanks have mostly been used on relatively low aspect ratio wings, which lends some credence to the magnitude of this problem.