TL;DR No, it won't die. Lizards/snakes are usually immune to their own venom.
Since I couldn't find a good answer to this recurring question on this site, I will try to summarize it here. I found more research on snakes than on lizards, but for now we will just assume that similar mechanisms can be found in lizards.
First, the question is how the venom kills the prey. What does it do in the body?
A common mechanism is a block of acetylcholine receptors at the neuromuscular junction. Simplified, these receptors activate muscle contraction upon a neuronal signal. When the receptor is blocked by the toxin, the neuronal signal cannot be transmitted to the muscle, which paralyzes the animal. Research has shown, that some snakes and lizards have evolved different acetylcholine receptors, that can still bind acetylcholine (and activate muscles), but not the toxins (source1, source2). 671
This makes them immune to the most common neurotoxins found in reptile venom.
Another mechanism is hemorrhagic activity of the toxin, which is a disruption of blood vessel integrity resulting in bleeding and death. This was linked to a proteolytic ("destroying proteins") function of venom compounds (source3, source4, source5). The blood serum of snakes can contain factors that block the hemorrhagic function of toxins, possibly by neutralizing the mode of action (inhibiting proteases). This could be specific antibodies/inhibitors or an adaptation of serum proteins (source6, source7, source8).
Further research has implicated phospholipidases (destroying phospholipids like used in cellular membranes) as the mode of action for snake venoms. Again, phospholipidase inhibitors can be found in the blood serum of the snakes themselves. There were also reports, that the expression of these inhibitors changes from younger to older snakes (source9, source10) and can be induced by venom injection (source11). It is therefore hypothesized that venomous snakes or lizards are repeatedly exposed to small amounts of their own venom, building up immunity over their life span.
Other modes of action include myotoxicity (destroying muscles), edema-formation or blood thinning. Again, it depends on the specific mechanism and how snakes and lizards have adapted to be immune to these.
Since similar toxins are found in several species, snakes and lizards are often not only immune to their own venom, but also to bites of other members of the same species or other related species. 112