Update after Xanathar's Guide to Everything
The currently accepted answer to this question opens with "That depends on whether sleeping gives you the Unconscious condition", which is correct. However, that answer was written in 2015. More than two years later Xanathar's Guide to Everything was published, and that reference book added optional rules for sleep that impinge on this answer, such that the answer to this question now crucially depends on whether these optional rules are in use.
With the optional rules from XGtE
While a creature sleeps, it is subjected to the unconscious condition.
Here, a sleeping character is unequivocally unconscious. And, per the PHB:
An unconscious creature is incapacitated (see the condition), can’t move or speak, and is unaware of its surroundings
And per the PHB rules on Concentration:
You lose concentration on a spell if you are incapacitated or if you die.
Thus, if the optional rules from XGtE are in play, sleeping results in the unconscious condition, which means you are incapacitated, which means you lose Concentration.
But with PHB / DMG Only
If the DM is not using XGtE, it is their call whether or not sleeping ends Concentration, because in this case sleeping does not equal the unconscious condition. Rather, sleeping is a state to which a DM may assign consequences at their discretion.
From the DMG "Using and Tracking Conditions" (p. 248 - emphases mine)
Various rules and features in the game are clear about when they apply a condition to a creature. You can also apply conditions on the fly. They're meant to be intuitive for you to do so. For example, if a character is in a state such as sleep, that lacks consciousness, you can say the character is unconscious. Or did a character just stumble onto the ground? He or she is now prone.
This passage does not say that sleeping characters per force have the unconscious condition. Rather, the point is that a DM can choose to apply the mechanical effects of conditions 'on the fly', that is, when they feel the situation as such warrants it. In the case of sleep, it is not that a character is unconscious, but rather that they lack consciousness and thus can be treated for some situations as if they were unconscious (and note that this is a different interpretation of the DMG quote than given in the accepted answer).
One of the consequences of the unconscious condition is that a creature 'is unaware of its surroundings' and thus cannot be woken up by sounds, lights, or perhaps shaking. This is pretty unreasonable in the case of normal sleep, so most DMs would likely not apply this aspect of unconsciousness to a sleeping creature, but they would be within RAW to do so (and note that even though XGtE says that sleeping creatures have the unconscious condition, it also includes rules for them waking up in response to external stimuli).
Another consequence of unconsciousness is that attack rolls against such a creature have advantage. This seems like a reasonable effect to apply to a sleeping creature, but it would again be up to the DM to decide whether or not this applies.
Whether or not sleeping means that you are incapacitated enough to end Concentration on spells is a third potential consequence of unconsciousness and is thus similarly left up to the DM to decide if they are working only with the official PHB / DMG rules.