I am not familiar with the concept of entropy for time series. I am looking for good reference papers and examples of use.
As a good starting point read this recent paper by Jing Chen:
For a special use of the entropy concept for forecasting the '87-crash read this paper:
(Although I tried to contact the authors to get the data to reproduce their findings, which they didn't send, it is still an enlightening read)
For a more popular exposition of the use of entropy in money management (key word 'Kelly formula') you should read this intelligent page turner by Poundstone: Fortunes Formula
EDIT: Quite an interesting paper is this one where Black-Scholes is derived through the use of concepts of relative entropy: http://www.mdpi.com/1099-4300/2/2/70/
Google for granger causality and its general version, transfer entropy, for a measure of whether a time series has a causal relationship with another (measured by calculating how much the conditional entropy of a time series decreases if we know another one, conditioned on everything else we know).
I have applied the concept of entropy and more specifically conditional entropy to spreading (ie, as a pricing model to get a sense for value) & execution decisions. It's good for everytime you're facing a problem of the sort, given X what is the probability density function of Y.
Also, the concept of mutual information which evaluates mutual dependence between two (random) variables can be useful in many applications. Again spreading comes to mind, risk management, etc
No papers on hand, but as usual the wiki is pretty good one going