Factors, whose only connection with the considered variables is that they influence only the dependent variable, in particular, have no connection with the independent variable, will not cause bias to your results so they don't have to be controlled. However, they could improve the precision. This paper is a good introduction; in particular, consider model eight therein.
But note how difficult it is to assess influence. E.g. in your example, it might very well be that the temperature is influencing the age because maybe older people will only run at lower temperatures. If, in addition, people run in general faster at lower temperatures, the temperature will be a confounder.
Because of the discussion in the comments, I would like to describe how you convince yourself whether there is a causal influence (causal effect) from temperature to age. Imagine you could create lots of experiments (marathons) and that you could arbitrarily set the temperature, i.e. decide about the marathon temperature without being influenced by anything else in the universe. But, while you remove all influence on temperature (removing all "incoming arrows"), you still allow the temperature to influence other variables as it did before (leave all "outgoing arrows" alone). That is an intervention. Then, would there be a stochastic dependence between the age and the temperature? I.e. would there be different probability distributions over age for different values of the temperature? I think so because older people will not run in high temperatures. That is what it means that temperature has a total causal effect on age in this scenario.
Furthermore, this would not work the other way around: If you intervened on the age of the runners, e.g. by simply forbidding older people to run, doing so would not change the temperature. I.e. age is not a cause for temperature in this scenario.
In the same way, intervening on the temperature would still show a dependency between temperature and runtime, while intervening on the runtime, e.g. by just stopping runners for an hour, would not change the temperature. Thus, the temperature is a cause for the runtime but runtime is not a cause for temperature.
Thus, the temperature is a confounder of age and runtime. And confounders are variables that you need to control.