Here's just a quick shot off the hip with a few thoughts and ideas.
In any case, it is difficult to give a reliable answer without having looked at the actual data in some depth: which approaches are likely to work and be practical will depend on what the data actually looks like, in particular how fast temperatures vary in each of the temperature series.
The first, and simplest, is the suggestion already made by G Cab, to use interpolation to fill in the gaps. If one point has more frequent measurements, that might be the one you'd do the interpolation on.
However, if the measurements are sufficiently far between that the temperature may have varied substantially during the period, this might not be a reliable approach. The problem is that the interpolated values will typically tend to be more stable than the real measurements: at least if they are some kind of weighted mean of the closest measurements.
2. Interpolation using time-series models
You can try to set up a time-series model to model how the temperatures vary. You can then assume you have measurements made at given points (which may well be random/irregular), and try to estimate the actual temperature.
On kind of method for doing this kind of estimation is the Kalman filter, although I guess you can find similar solutions (which may have different names).
In many respects, this is similar to the interpolation, except you have a more advanced method for doing the interpolation. You can also take into account that accuracy of the estimates so that the points in time when you have accurate estimates are given greater emphasis in the computation of the correlation.
3. Correlated time-series models
You can go one step further with the time-series models and assume that there are two correlated time-series with measurements taken at arbitrary time points, fit the time-series, and try to estimate the correlation based on the model.
This is similar to point 2 except instead of actually interpolating the temperatures and computing correlations on the interpolated values, you fit a more complex model and estimate the correlation between the two models.
In any case
In any case, you should try to make different models for temperature series and for the timing of measurements in order to generate random data for which you know the actual correlation, and then test your methods on those. By "model", I here mean just some method of generating the temperature series (for all time points so you can compute the actual correlation!) and then sampling the random measurement points to see how well you can estimate the actual correlation.