Are (or were) there any operational aircraft that don't have horizontal and vertical stabilizers or canards? If so, what's the effect on how they fly?
Depending on how you classify the control surfaces, the B-2 Spirit does not have a horizontal stab, since it is a "flying wing" design. As you have changed the question, the other part of the answer is that the flying wings have been able to get by without vertical stabilizers as well, at least in the case of the B-2. It uses both spoilers ("split brake rudders") and thrust differential to accommodate the lack of a vertical stabilizer.
Northrop had investigated several means of applying directional control that would least infringe on the aircraft's radar profile, eventually settling on a combination of split brake-rudders and differential thrust ~ Sweetman, Bill. "Lockheed Stealth" (2005) ISBN 0-7603-1940-5., p. 73
(And as with flight controls in general on the on B-2, that's helped a lot by its fly by wire flight control system).
As the Germans discovered in their flying wing aircraft (Ho229), and the USAF in the YB-35 and YB-49 programs, flying wings have inherent instability, even though some early biplane "flying wing designs" were stable enough to fly. (Dunne).
The B-2 has a fly by wire system to retain stability and handling qualities so that it does not have to make tradeoffs (loss) in performance that earlier flying wing designs did in order to make the design "stable enough."
What the elevator/horizontal stab do for conventional aircraft is allow for longitudinal control (pitch control). If you don't choose to use that conventional method, the "effect" is that your design must find another way to establish longitudinal control and stability or the aircraft risks being unstable in that axis.
Likewise with vertical stabilizers, your design has to address stability in the yaw axis if you want to make it without a control surface acting in that axis.
Based on the wording of your question, the Flying Wing design would qualify. These aircraft do not have proper "tail" stabiliser sets and they still fly.
Among the earliest aircraft to fly were the models of Alphonse Pénaud, and they flew without a vertical tail. The rubberband-powered pusher propeller was sufficient for stabilization. Admittedly, they did not need to maneuver.
Pénaud's Planophore (picture source). The spinning propeller replaced the vertical tail.
After Friedrich Ahlborn published a study on the aerodynamic qualities of the Alsomitra seed, Igo Etrich and Franz Wels designed a glider inspired by the seed's shape. This flew in 1906 and was the first man-carrying flying wing. As a glider, it could not rely on a pusher propeller for stabilization; instead it copied the sweptback outer wing of the Alsomitra seed with its upswung trailing edge.
Flying wings use reflex airfoils and/or wing sweep in combination with reduced incidence in the outer part of the wing (washout) for stabilization. Yaw stability is provided by sweep and pitch stability by the reduced incidence in the outer/rear part of the wing. In general, avoiding tail surfaces altogether can still result in stable aircraft but reduces damping a lot. The result is a flyable but problematic design.
No self-respecting hang glider designer would think twice about adding tail surfaces (picture source).
A third way of avoiding a conventional tail is the V-tail, which was invented in 1930 by the Polish engineer Jerzy Rudlicki.
Ultraflight Lazard ultralight with inverted V-tail (picture source)
In the 1940s and 1950s, roll, pitch and yaw control was demonstrated through variable wing geometry, the aim being to eliminate the drag and normally negative lift of the tail.
When variable geometry wings were re-invented in the 1960s, the original purpose had been forgotten or discounted, and variable geometry was restricted to matching the wing's characteristics to different speed flight regimes, pitch and yaw control being performed by conventional tail surfaces.
As they had both the weight of wing hinges and actuators and the disadvantages of tail control surfaces, they were probably the worst of both worlds.
The experiments were restricted to remote controlled prototypes, to prove the principle before developing manned aircraft (their designer was distressed by the loss of life from some earlier inventions). Apparently, money issues and politics stopped further development at the time in the UK, though this page suggests Nasa studied the idea, finding some difficulties including unexpected drag. Whether those difficulties could be solved now ... I've no idea. I'm no expert, just an observer, so I'd be interested to hear any more about these ideas.
Pity, the Swallow (source here) would have been a nice looking machine.
Sources: Life of Barnes Wallis