Do Na or Li need heat in order to react with Cl₂?
From personal experience, Na(s) and Cl2(g) react at room temperature without added energy. Li(s) and Cl2(g) should do so also.

Consider the following heat of formation data harvested from the NIST Chemistry Webbook:

For NaCl:
$$\ce{Na(s) + } \frac 1 2\ce{Cl2(g)-&gt;NaCl(s)}\quad \Delta H_f^o=-411.12 \; kJ/mol $$

For LiCl:
$$\ce{Li(s) + } \frac 1 2\ce{Cl2(g)-&gt;LiCl(s)}\quad \Delta H_f^o=-408.27 \; kJ/mol $$

These data suggest that both reactions should occur without added heat. However, the rate of reaction is not determinable from these data. The reaction between Li(s) and Cl2(g) will likely be slower.