The answer is yes if your domain has compact embedding $H^1(\Omega)\to L^2(\Omega)$ (that is, for any regular enough domain), or you consider $H^1_0(\Omega)$, but no in general. Counterexample is as follows:

Let $\Omega = (\frac{1}{2}, 1)\cup (\frac{1}{8}, \frac{1}{4})\cup (\frac{1}{32}, \frac{1}{16})\cup \ldots = I_1\cup I_2\cup \ldots \subset \mathbb{R}$ and consider functions $f_n = \frac{1}{\sqrt{|I_n|}}\chi_{I_n}$. They all are bounded in $H^1(\Omega)$ (indeed, they has norm $1$), but you can't find a compact $K\subset \Omega$ such that $K\cap I_n \neq \varnothing$ for all $n$ because such a $K$ has to contain $0$ which is not in $\Omega$.

If you do not like nonconnected examples, consider $\Omega_1 = \Omega \times [\frac{1}{2}, 1) \cup (0, 1)\times (0, \frac{1}{2})$ (it looks like a comb), choose a smooth function $g$ with support in $(\frac{3}{5}, \frac{4}{5})$ and consider $g_n(x, y) = f_n(x)g(y)$.

On the other hand, if you have a compact embedding then the answer is yes. First of all let us consider a sequence of compact sets $K_n\subset \Omega$ such that $K_n \subset Int(K_{n+1})$ and $\bigcup K_n = \Omega$ (such a sequence of compacts exists for any manifold and a fortiori for any open subset of $\mathbb{R}^d$). Assume that there exists $\varepsilon > 0$ such that for any $n$ there exists $u_n\in H^1(\Omega)$, $||u_n||_{H^1(\Omega)} = 1$ such that $||u_n||_{L^2(\Omega \backslash K_n)} \ge \varepsilon$. By compactness of embedding (passing to a subsequence, but for notational simplicity I will assume that it is the same sequence) we may assume that $u_n\to u$ in $L^2(\Omega)$. But functions $v_n = |u|^2 \chi_{\Omega \backslash K_n}$ has a common $L^1$-majorant $|u|^2$ and converges pointwise to $0$. Therefore $||v_n||_{L^2(\Omega)} < \varepsilon$ for some $n$ by Lebesgue Theorem. On the other hand $w_m = u_m \chi_{\Omega \backslash K_n}$ converges to $v_n$ in $L^2(\Omega)$. Thus, for big enough $m$ $||w_m||_{L^2(\Omega)} < \varepsilon$. But on the other hand $||w_m||_{L^2(\Omega)} \ge ||u_m||_{L^2(\Omega \backslash K_m)}\ge \varepsilon$ for $m > n$ -- a contradiction.