In this work, we address the question of the largest rate of linear subcodes of Reed-Muller (RM) codes, all of whose codewords respect a runlength-limited (RLL) constraint. Our interest is in the $(d,\infty)$-RLL constraint, which mandates that every pair of successive $1$s be separated by at least $d$ $0$s. Consider any sequence $\{{\mathcal{C}_m}\}_{m\geq 1}$ of RM codes with increasing blocklength, whose rates approach $R$, in the limit as the blocklength goes to infinity. We show that for any linear $(d,\infty)$-RLL subcode, $\hat{\mathcal{C}}_m$, of the code $\mathcal{C}_m$, it holds that the rate of $\hat{\mathcal{C}}_m$ is at most $\frac{R}{d+1}$, in the limit as the blocklength goes to infinity. We also consider scenarios where the coordinates of the RM codes are not ordered according to the standard lexicographic ordering, and derive rate upper bounds for linear $(d,\infty)$-RLL subcodes, in those cases as well. Next, for the setting of a $(d,\infty)$-RLL input-constrained binary memoryless symmetric (BMS) channel, we devise a new coding scheme, based on cosets of RM codes. Again, in the limit of blocklength going to infinity, this code outperforms any linear subcode of an RM code, in terms of rate, for low noise regimes of the channel.