  We present the results of {\it Chandra} observations of two non-accreting
millisecond pulsars, PSRs J1640$+$2224 (J1640) and J1709$+$2313 (J1709), with
low inferred magnetic fields and spin-down rates in order to constrain their
surface temperatures, obtain limits on the amplitude of unstable $r$-modes in
them, and make comparisons with similar limits obtained for a sample of
accreting low-mass X-ray binary (LMXB) neutron stars. We detect both pulsars in
the X-ray band for the first time. They are faint, with inferred soft X-ray
fluxes ($0.3-3$ keV) of $\approx$ $6\times10^{-15}$ and $3\times 10^{-15}$ erg
cm$^{-2}$ s$^{-1}$ for J1640 and J1709, respectively. Spectral analysis
assuming hydrogen atmosphere emission gives global effective temperature upper
limits ($90\%$ confidence) of $3.3 - 4.3 \times 10^5$ K for J1640 and $3.6 -
4.7 \times 10^5$ K for J1709, where the low end of the range corresponds to
canonical neutron stars ($M=1.4 M_{\odot}$), and the upper end corresponds to
higher-mass stars ($M=2.21 M_{\odot}$). Under the assumption that $r$-mode
heating provides the thermal support, we obtain dimensionless $r$-mode
amplitude upper limits of $3.2 - 4.8 \times 10^{-8}$ and $1.8 - 2.8 \times
10^{-7}$ for J1640 and J1709, respectively, where again the low end of the
range corresponds to lower-mass, canonical neutron stars ($M=1.4 M_{\odot}$).
These limits are about an order of magnitude lower than those we derived
previously for a sample of LMXBs, except for the accreting millisecond X-ray
pulsar (AMXP) SAX J1808.4$-$3658, which has a comparable amplitude limit to
J1640 and J1709.
