Technically, a proper burst can't usually be centered on the creature that created the burst...
When a spell or effect affects an area that's a burst
Regardless of the shape of the area, you select the point where the spell originates, but otherwise you don’t control which creatures or objects the spell affects. The point of origin of a spell is always a grid intersection.... Most spells that affect an area function as a burst, an emanation, or a spread. In each case, you select the spell’s point of origin and measure its effect from that point.
A burst spell affects whatever it catches in its area, even including creatures that you can’t see. It can’t affect creatures with total cover from its point of origin (in other words, its effects don’t extend around corners). The default shape for a burst effect is a sphere, but some burst spells are specifically described as cone-shaped. A burst’s area defines how far from the point of origin the spell’s effect extends.
Emphasis mine. So when a creature casts, for example, the 2nd-level Clr spell sound burst [evoc] (PH 281) with its area of a 10-ft. radius spread or the 8th-level Sor/Wiz spell sunburst [evoc] (PH 289) with its area of an 80-ft. radius burst, the crosshairs at which the area begins can be adjacent to the caster, but the point at which the area begins can't be a square the caster occupies. (Many games such as GURPS and the Hero System—even some d20 games like either edition of Spycraft—do count actual hexes or squares instead of counting crosshairs to crosshairs when dealing with burst-like area, so Dungeons and Dragons 3.5 is exceptional in this case.)
...But everything described as or named a burst isn't, technically, a burst
Sometimes names and descriptions of effects in Dungeons and Dragons demonstrate a shocking lack of imagination. Lots of things that are called bursts don't conform to the technical definition of a burst. Seriously, the 1st-level psionic power burst [psychoportation] (XPH 81-2), for example, isn't a burst at all! And the 0-level Sor/Wiz spell flare [evoc] (PH 232) says Effect: Burst of light, yet that burst of light affects but a lone creature.
The dread necromancer's supernatural ability negative energy burst is one of these fancifully named bursts—not a technical burst—, its specific description overriding the general rules of bursts with its details describing exactly how it functions:
Beginning at 3rd level, a dread necromancer gains the ability to emit a burst of negative energy from her body, harming living creatures within 5 feet of her. (Heroes of Horror 86)
This effect affects creatures in and adjacent to the dread necromancer's space but doesn't affect the dread necromancer himself because he's emitting the effect. Would it be easier to adjudicate were it phrased like the 9th-level Sor/Wiz spell wail of the banshee [necro] (PH 298)? Sure, and such a description would also make it clear how the special ability charnel burst interacts with obstacles, yet negative energy burst's phrasing is clear and playable enough.
(Note that the special ability negative energy burst is hardly unique in being a burst centered on the burst's creator, especially taking into account psionic powers. For example, the 3rd-level psion/wilder power energy burst [psychokinesis] (XPH 101) is a burst centered on the manifester yet dealing no damage to the manifester, while the 6th-level psion/wilder power breath of the black dragon (XPH 81) is a cone-shaped burst centered on the manifester without such an exception! It would've been kind of the developers to've included rules for adjudicating such effects, however. In particular, emanations—especially the effect created by the 6th-level Sor/Wiz spell antimagic field [abjur] (PH 200)—would've benefited greatly from only a few more words of explanation and a diagram.)
In sum, a dread necromancer's negative energy burst does not affect the dread necromancer...
Thus, while it is possible to catch oneself in the area of one's own burst—like in a fireball spell's Area: 20-ft.-radius spread (which, by the way, has been happening for literal decades) or whatever—, it's not possible for a dread necromancer to catch himself in the area of his own negative energy burst. Your player's tomb-tainted dread necromancer is unaffected by his own special ability negative energy burst, but, if it's any consolation, he can still touch himself as a standard action to benefit from his supernatural ability charnel touch.
...But a house rule saying that it can is totally okay
Even if a house rule allows the tomb-tainted dread necromancer to take a standard action 1/day (+1 time/day at levels 8, 13, and 18) to use the supernatural ability negative energy burst to deal 1d4 points of negative energy damage/level to creatures adjacent to his space and to himself, game balance won't be upset. Taking a standard action to heal a comparatively minor amount of damage to himself and some minions (while possibly endangering his living allies) when that standard action could've been spent casting spells that kill people? That's a choice a dread necromancer can be allowed to make without fear of him dominating the game.