To find the sum, we look at the first few powers of 2 modulo 7: \begin{align*} 2^0 &\equiv 1, \\ 2^1 &\equiv 2, \\ 2^2 &\equiv 4, \\ 2^3 &\equiv 8 \equiv 1 \pmod{7} \end{align*}Since $ 2^3 \equiv 1 \pmod{7} $, the powers of 2 modulo 7 repeat in cycles of 3. Therefore, \begin{align*} &1 + 2 + 2^2 + 2^3 + \dots + 2^{100} \\ &\quad\equiv 1 + 2 + 4 + 1 + 2 + 4 + \dots + 1 + 2 + 4 + 1 + 2 \\ &\quad\equiv (1 + 2 + 4) + (1 + 2 + 4) + \dots + (1 + 2 + 4) + 1 + 2 \\ &\quad\equiv \boxed{3} \pmod{7}.\end{align*}