For a positive integer $n$, let $R_n$ be the set of integer lattice points $(x, y)$ such that
- $0 \leq x < 2n$
- $0 \leq y < 4n$
- $x \leq y$
- $y \leq 5n - x$
- $y \leq x + 3n$,
and let $L_n = \{(-x, y) \mid (x, y) \in R_n\}$. Define the $n$th heart board to be $L_n \cup R_n$. The first four heart boards look like this:
Can heart boards ever be tiled with dominoes?
The $n$th heart board seems to have size $10n^2 + 3n - 3$. This is even iff $n$ is odd, so only the odd heart boards have any hope to be tiled. With a computer search I've verified that no heart board with $n \leq 20$ can be tiled using dominoes. I don't see how to extend any "forcing" argument to arbitrary integers, but I suspect that there is some nice pattern. (Any possible tiling seems to only use "upright" dominoes, for example.)
No heart boards can be tiled with dominoes.
Gregory Puleo's suggestion turns into an excellent proof.
"Color" the heart with $+1$ and $-1$ so that the spaces alternate signs. Every domino then covers exactly one $+1$ and one $-1$, so the heart board will not be tileable if there are more $+1$'s than $-1$'s, or vice versa. It suffices to consider the heart a single column at a time.
Fix some $x \in \{0, 1, 2, \dots, 2n - 1\}$. What is the height of the $x$th column? Using the given constraints, it is exactly
$$\min(5n - x, x + 3n, 4n) - x + \delta,$$
where $\delta = 0$ if the minimum is $4n$, and $1$ otherwise. In the first case for the minimum, we get
$$5n - x - x + 1 = 5n - 2x + 1.$$
This is even if $n$ is odd, so the "color difference" is $0$. In the second case for the minimum, we get
$$x + 3n - x + 1 = 3n + 1,$$
which is even if $n$ is odd, so the color difference is again $0$. In the third case we get
$$4n - x,$$
but the conditions $4n \leq 5n - x$ and $4n \leq x + 3n$ happen to imply $x = n$, so the color difference is actually $3n$, and this occurs in only one column. This is odd if $n$ is odd, so there is exactly one more plus sign than minus sign in this column (or the other way around).
This all means that the odd heart boards have a color difference of exactly two, one from column $x = n$, and one from column $x = -n$, so they cannot be tiled by dominoes. The even heart boards have an odd number of spaces, so they obviously cannot be tiled with dominoes.