Is the Dirac delta (impulse) signal a power signal or an energy signal?
[Added a reference on Schwartz's impossibility theorem for products of distribution] 

The continuous Dirac delta  $\delta$ is not considered a true function or signal, but a distribution. From its wikipedia page:


  The delta function can also be defined in the sense of distributions
  exactly as above in the one-dimensional case.[25] However, despite
  widespread use in engineering contexts, (2) should be manipulated with
  care, since the product of distributions can only be defined under
  quite narrow circumstances.


It can be defined such that, for any function $f$ satisfying some important properties, and for $a\in \mathbb{R}$:

$$ \int f(t)\delta(t-a)dt=f(a).$$

From the best of my knowledge, those important properties are not satisfied by $\delta$, so one cannot directly replaced $f$ by $\delta$ and get a meaningful result. As far as know, the product of two Dirac distributions is not well defined, unless one talks about $n$-dimensional versions, or so-called "formal" manipulations as used in physics for instance, or more complicated maths. 
A short account of Simplified production of Dirac delta function identities is provided by Nicholas Wheeler. If one wants to dig deeper, I'd suggest The Colombeau theory of generalized functions, by Ta Ngoc Tri, 2005:


  Soon after the introduction of his own theory, L. Schwartz published a
  paper in which he showed an impossibility result (see [Sch54]) about
  the product of two arbitrary distributions.


One result is  Schwartz impossibility result. It (somehow) says that if one wants to encompass the derivative of continuously differentiable functions whle keeping  Leibniz's rule of derivation, then one get $\delta^2(|x|)=0$.

However, from an informal point of view, sometimes used in DSP (and in physics), this "product" is, as far as I know, neither energy nor power. 
From a logical point of view though, if it does not exist, one could affect this "product" a lot of properties...

Some related posts:


What is product of delta function with itself?
What is the square root of the Dirac Delta Function?
Products and compositions with the Dirac delta function, C. K. Raju, Journal of Physics A: Mathematical and General, Volume 15, Number 2