I often see beginner sheet music with "rhythmically" or "with a beat" in the top left hand corner. I don't understand how this is supposed to be interpreted. Isn't all music with a time signature supposed to have a beat? What is an example of music which is not meant to be played rhythmically?
This is certainly not a marking confined to music for beginners. It is a perfectly legitimate marking; for instance, in Italian the marking is ritmico.
Any passage of music can be played making a greater or lesser emphasis of the rhythm of the music. It is possibly easiest to think about the effect of not playing rhythmically (without rhythmic accentuation), in other words, making the change between notes as smooth as possible, with little attack, and instead as a flowing stream of notes. If one is playing rhythmically, on the other hand, the rhythm of the notes is made more obvious. There are a number of ways to do this, one could: play notes in a more detached or even staccato manner (this would be instrument specific: changing bow on a string instrument; tongueing on a wind instrument, for instance); play each note with a more distinct attack at the beginning; play with a louder, more forceful sound generally (rather than with a soft, mellow, gentle sound).
What is an example of music which is not meant to be played rhythmically?
More seriously, they're calling for the beats to be emphasised, much like most marches and dances. Other kinds of music may indeed have a beat as a rhythmic foundation, but may not particularly emphasise it, may at times even work against it.
There are plenty of examples of music that doesn't particularly emphasise the beat. I can think of three right off the top of my head: Vergangenes and Farben from Schoenberg's Five Pieces for Orchestra, and the opening movement of the Music for Strings, Percussion and Celesta by Bartók.