Can individual odds ratios be added to get one pooled odds ratio to compare to a reference group? I have the admission for 4 different hospitals (coded as binary) which are regressed on a list of independent variables.
The multinomial logistic provides the odds ratio (O.R.) for independent variable for each hospital and then compare this to a reference hospital.
Can I add each of the O.R. from each independent variable and get a combined/pooled O.R. for each hospital and then compare this pooled O.R. to a reference hospital and to the individual hospitals? 
 A: No, unfortunately you can't simply add the odds ratios.  If you look up any text or reputable source on logistic regression you'll find formulas with which to estimate either the logit or the odds or the probability (depending on what's most useful to you) of admission to a given hospital, given certain values on the predictors.
I agree with @Simon and would add that for multinomial regression you must have a single variable that takes 4 different values depending on the hospital--not 4 binary variables.
A: As far as I know, there is no situation in which you can add odds ratios and get something sensible. There are SOME situations (but not yours) where you can add the parameter estimates and then recalculate odds ratios.  
To get what you seem to want is pretty easy: Recode the variable as a single binary variable 0 for "other hospital" and 1 for "reference hospital". But be careful because this gets into the whole area of fishing for significant results, post-hoc vs. a priori tests, and so on.
A: Not only can you not add odds ratios, you'd be hard pressed to find any kind of ratios that are not proportions that can be added.  The simplest way to see that the math doesn't work for adding ratios is that the original direction of the effect is lost.  Even a negative regression coefficient gives rise to an anti-logged value that is positive, and adding this positive is as if the effect were harmful.
A: If they are part of the same study and the variable are independent I think ORs can be combined responsibly. Considering:
$$
Odds = exp( \beta_0x_0 + \beta_1x_1 + ... + \beta_nx_n )
$$
Since, $OR = e^{\beta}$, we can retrieve $\beta$ by with the natural log: $ln(OR = e^{\beta}) \Rightarrow ln(OR) = \beta$. Since $x$ is binary in this case we are almost able to produce the Odds. If we know the base expected odds for the outcome then we are there:
$$
Odds = exp( ln(OR_0)x_0 + ln(OR_1)x_1 + ... + ln(OR_n)x_n )
$$
Does this make sense?
