First, note that your terminology is inconsistent. Here I take it that you have one variable (not several) consisting of a fixed number of categories and you are concerned with how categories with zero frequency or probability (not value) are handled.
Your $H$ is evidently $\sum p_i\ \text{log}_2\ (1/p_i)$ for probabilities or proportions $p_i$. The base used for logarithms does not affect any key principle here so we can think that we are summing terms $p_i\ \text{log}\ (1/p_i) = -p_i\ \text{log}\ p_i$.
The counter-argument to your worry is that entropy does take into account categories that have zero probability; it is just that they contribute zero to the entropy given that a strong convention that $-0\ \text{log}\ 0$ is evaluated as 0. A more informal version of the same argument is that the diversity or non-uniformity of what you do have in your collection is unaffected by what you don't have. If I have 10 elephants, spelling out that I have 0 giraffes or do not have any giraffes is incidental: what I have are 10 elephants. Any other statement about 0 frequencies adds no information (literally).
The same question of how to handle zero proportions arises with any measure. An alternative to entropy is based on squaring probabilities $\sum p_i^2$ and with such measures there is the same consequence that any $p_i$ that is 0 makes no difference to the sum.
You touch on a much more general issue of what can be inferred about a distribution from a summary measure. But any single summary measure is a irreversible reduction; you can't go back to the distribution unequivocally. This is on all fours with the point made in elementary statistics that a mean or correlation can reflect quite different data.
I suspect that the main issue here is that you are seeking a way to make entropy more intuitive and that is a legitimate concern. An easy way is to talk in terms of the "numbers equivalent". Calculate $2^H$ for your examples and you recover 5 for 10,10,10,10,10 and 1 for 10,0,0,0,0, which have the interpretation as the equivalent number of (equally common) categories that are present. For other examples, the result will be a non-integer, which is reasonable. For bases 10 or $e$, use $10^H$ or $\exp(H)$ to get the numbers equivalent.
P.S. I try to avoid asserting that something is meaningless unless I am totally sure that it is. I have found too often that I just didn't understand the argument.
EDIT 2016: If you know that (e.g.) 4 and only 4 categories are possible in principle, but only 3 occur, then that's pertinent information. Sometimes you know this: e.g. if cards can be $\{$spades, hearts, clubs, diamonds$\}$ and only some of those kinds occur, that's something to cite.
A measure of diversity that does take zeros into consideration, and is affected by whether zeros occur, has various names (e.g. dissimilarity index) and has general form $(1/2) \sum_{i=1}^S | p_i - q_i | =: D$ (say). Here $p_i$ is the observed proportion of category $i$ and $q_i$ is the proportion in a reference distribution, e.g. equal probabilities $q_i = 1/S$. Then the minimum occurs when the observed distribution is identical to the reference distribution and then $D = 0$. The maximum occurs when one proportion $p_i$ is $1$ and the others all zero. The achievable maximum depends on the number of categories $S$, which after all is part of the information. The concrete interpretation of $D$ is the minimum proportion that would need to change categories to reproduce the reference distribution.
Another example of a reference distribution would be the national distribution of different socio-economic classes or ethnic categories. Then $D = 0$ might mean that a local or regional community is a microcosm of the national and otherwise $D$ measures departure from that in some direction.
