Endothermic or Exothermic
Because $\Delta H$ is temperature-, pressure-, and concentration-dependent, it's certainly theoretically possible for $\Delta H$ for a chemical reaction to change sign in response to changes in any of those intensive properties.  

I don't have an example of such a chemical reaction handy.  However, there are direct experimental measurements showing enthalpies of solution that change sign in response to both temperature and concentration changes: 

For instance, the enthalpy of solution of polystyrene in decalin has been shown to change from ~ –10 J/g at ~20 C to ~ +6 J/g at ~100 C (the ~'s were needed because I'm eyeballing these from a plot).  Reference:

Tager, A. A., et al. "The temperature and concentration dependence of enthalpy, free energy and entropy of mixing polystyrene solutions with an upper and lower critical temperature of mixing." Polymer Science USSR 12.6 (1970): 1497-1511.

And the enthalpy of solution of $\ce{LiNO_3}$ has been shown to change from negative to positive as the concentration increases.  Reference:

Wolf, G., V. Vacek, and V. Pekarek. "Enthalpies of solution and of crystallization of lithium nitrate and of lithium nitrate trihydrate in water at 25° C." Journal of Solution Chemistry 19.10 (1990): 1029-1039.