Are there stars that wouldn't look white to the naked eye?

The difficulty of seeing star's colours is that our colour vision isn't sensitive to low light - at night we are mostly monochromatic.

Having said that, some of the brightest stars have enough light to show colours. If you look in Orion the star Betelgeuse (top left) looks red and Rigel (low right) looks blue(ish) in good conditions.

This is actually a really interesting question and the answer is, indeed, a little bit complicated, and combines what has been suggested in the other comments but also goes somewhat further as well.

First off - this will depend strongly on what distance you are imagining viewing the stars from. As mentioned, most stars, when viewed from Earth, are too dim to notice their colors: this is because your eyes are generally not sensitive to color at such low levels of stimulation by light. In particular, your eyes contain two different types of photoreceptor cells called rods and cones, where the rods are considerably more sensitive to low levels than the cones are, but only the cones are used for color perception (they have multiple wavelength sensitivity patterns, while rods have only one). For most stars, only enough light reaches your eye to stimulate the rods but, as mentioned, some of the brightest stars do just give off enough to stimulate the cones.

But, if you use some type of passive optical aid, such as a telescope, which will magnify the stars and so increase the amount of light that enters your eyes from them, and you may then be able to see more of their colors. Effectively, using a telescope "decreases the distance" to the star by an amount equal to the telescope's magnification - so if you are seeing it at 100x magnification, it is as though the star were 100x closer, provided of course you do not hit the telescope's diffraction limit.

However, there's another angle to this and that's that if you are to imagine actually being in the star system with the star (i.e. at planetary distance), then actually if you looked straight at any star, it would, indeed, appear white in terms of its observable surface color, no matter the temperature, although the incident light (and light seen in the "starburst"/halo it will create in your eyes due to scattering) may appear a color different from white if its temperature is significantly higher (you would want to have special shielding) or significantly lower than that of the Sun. This is not due to physics, but rather the nature of sensors such as your eyes and cameras. It is actually an effect you can observe with looking at hot metalwork in a blacksmithing shop or steel foundry. Molten iron emerging from a blast furnace is at a temperature of about 2000 K, which is cooler than even the coolest "Type M" stars (begins around 2600 K for objects hot and massive enough to actually be proper "stars", i.e. with nuclear fusion reactions persisting at their core), and yet when you look at it with your eyes, it appears whitish on the surface, but casts a heavily reddish hue on everything else around it. I believe this effect is due to the fact the light is so intense that it effectively "maxes out" the sensors across their full spectrum despite that it contains much more red than other colors, because even with that, the remaining colors are still extremely intensely emitted. At least this appears to be the case with cameras, since if you take a picture of that same molten iron but with the camera exposure stopped down dramatically (short shutter, closed iris) you will see it comes out the expected red color on its surface. Whether something similar is going on with the human eye or not is a question in psychophysics and the biology of vision for which I do not know the definitive answer, just that the effect is present and observable.

Thus the same would be expected to be the case as well with the stars. To get a visibly red surface that does not overwhelm your eyes, you will need a much cooler surface temperature, say around 1000 K, which is about that of the "red hot" stove burners on an electric stove turned up to maximum. This temperature is only attained for brown dwarf substellar objects which, as the name implies, are not stars: they do not carry out nuclear fusion, at least of hydrogen ($^1\mathrm{H}$) or heavier elements (deuterium fusion is possible at the beginning of their existences).

However, given that it would be inadvisable to stare into any stars from close distance for a protracted period of time with an unprotected eye for the same reasons as the Sun, you would probably want some heavy filtering goggles in front from these distances, and if you did that, and they were uniform in filtering across the visible spectrum, I believe you would see the "expected" surface color which, by the way, is not quite what you'd get looking at the names typically given to the spectral classes: type G stars, like the Sun, are actually white stars, not yellow. Real yellow is more like type K, and M is orange. Type F is bright white, A and above are bluer, up to welding arc-like hues at type O which has the surface temperatures equivalent to an electric arc, and better best be kept far enough away from that it is not larger in appearance to you in terms of angular size than exactly that, for how big something looks at a given temperature is also directly proportional, thanks to the coincidence of geometric laws, to how much light and heat you are receiving from it, as well as in this case dangerous UV radiation.

There are stars that are visibly not white even from Earth. Sirius, the brightest star in the sky (except the sun), has a perceivable blue tint, and Betelgeuse is noticeably red. That's just the beginning of answering your question though.

In general, hotter stars are more blue and cooler stars are more red. The sun is "just right" in its temperature to emit what is a very white spectrum. Sirius, mentioned above, is a much hotter star than the sun, and Betelgeuse is cooler. In fact, most of the stars in the galaxy are "very cool" (still thousands of degrees) small stars that would appear very red if you could see them. However, they're also quite faint, so they can't be seen by the naked eye from very far away and that's why the naked eye night sky is not dominated by them.