is it reasonable to consider Horse Colic dataset as a binary classification problem? I am learning Horse Colic dataset.

Well documented attributes; 368 instances with 28 attributes (continuous, discrete, and nominal); 30% missing values

there are 3 type of outcomes, 'lived', 'died', 'was euthanized'.

since 'was euthanized' is another form of 'died', is it reasonable to consider this problem as a binary classification problem?
It depends on your goal.

If you're interested in the horse's status then this is a multiclass classification problem.

Or if you're only concerned if a horse died and not concerned about how it died, then you can treat it as a binary problem (after grouping 'euthanised' with 'died').