Remove tick labels, but retain tick marks in RegionPlot (and related functions)

FrameTicks

Consider FrameTicks in the example below:

RegionPlot[Sin[x] Sin[y] > 1/4, {x, -1, 1}, {y, -1, 1},
  FrameTicks -> {Table[{k, "", {If[Mod[k, .5] == 0, .02, .01], 0}}, {k, -1, 1, .1}], 
                 Table[{k, "", {If[Mod[k, .5] == 0, .02, .01], 0}}, {k, -1, 1, .2}]}]

In Table[{k, "", {If[Mod[k, .5] == 0, .02, .01], 0}}, {k, -1, 1, .1}] ,

k is the location of the tick

"" is the label

If... determines whether to use a major (.02) or minor (.01) tick on the inside of the respective axis.

0 indicates that the tick should not appear on the outside of the axis.

Note that the first Table sets the horizontal edges of the frame (with ticks every .1 units); the second Table sets the vertical edges of the frame (with ticks every .2 units).

My preferred solution is playing with FontColor or FontOpacity as in R.M.`s answer, or define your own ticks as in David's answer.

Another alternative is to change the labels to blank in FrameTicks. Since FrameTicks->Automatic saves a lot manual effort (and it uses the built-in FindDivisions for selecting ticks), sometimes it may be more convenient to transform the automatic frameticks data:

 rp = RegionPlot[x^2 + y^3 < 2 && x + y < 1, {x, -2, 2}, {y, -2, 2}, 
 FrameTicks -> Automatic]

First extract the frameticks information and change the labels to blank:

 newticks = Last@First[AbsoluteOptions[rp, FrameTicks]];
 newticks[[All, All, 2]] = "";

Then replot or useShow with the new frameticks:

 Show[rp, FrameTicks -> newticks]

to get:

I don't believe this has been mentioned yet:

RegionPlot[1 < Abs[x + I y] < 2, {x, -2, 2}, {y, -2, 2}, ImagePadding -> 1]

ImagePadding
is an option for graphics functions that specifies what absolute extra padding should be left for extended objects such as thick lines and annotations such as tick and axis labels.

You can use the internal, undocumented function Charting`ScaledFrameTicks to do this (Charting`ScaledFrameTicks produces unlabeled ticks, while Charting`ScaledTicks produces labeled ticks):

RegionPlot[
    Sin[x y]>0,
    {x,-1,1},
    {y,-1,1},
    FrameTicks->{
        {Charting`ScaledFrameTicks[{Identity,Identity}], Automatic},
        {Charting`ScaledFrameTicks[{Identity,Identity}], Automatic}
    }
]