Let's think about regular linear regression, and to make it concrete, let's say we are trying to predict height of people. When you regress heights against just an intercept term and no predictors, the intercept term will be be the height averaged over all the people in your sample. Lets call this term $\beta_0^{\text{no predictor}}$
Now, we want to add a predictor for sex, so we create and indicator variable that takes a 0 when the sampled person is male and 1 when the person is a female. When we regress against this model, we will get an estimates for an intercept term, $\beta_0^{\text{male reference}}$ and coefficent of the sex variable $\beta_1^{\text{male reference}}$. The estimated intercept is no longer the average height of everybody, but the average height of males, the coefficient of the sex variable is the difference in the average height between males and females.
Consider if we decided to code our indicator variable differently, so that the sex variable took the value 0 if the person was a female and 1 if the person was a male, in this specification of the model we get the estimates of the intercept and coefficient $\beta_0^{\text{female reference}}, \beta_1^{\text{female reference}}$. Now $\beta_0^{\text{female reference}}$, the intercept term, is the average height of females, and the coefficient is the difference in average height between females and males. So
$$
\begin{align}
\beta_1^{\text{male reference}} &= -\beta_1^{\text{female reference}}\\
\beta_0^{\text{male reference}} + \beta_1^{\text{male reference}} &= \beta_0^{\text{female reference}}\\
\beta_0^{\text{female reference}} + \beta_1^{\text{female reference}} &= \beta_0^{\text{male reference}}
\end{align}
$$
So, by changing how we coded the indicator variable we changed both the value of the intercept term the coefficient term, and this is exactly what we should want. When we have a multivalue indicator, you will see the same kinds of changes as you specify difference reference levels, i.e. when the indicators take on the value of 0.
In the binary indicator case the p-value of the $\beta_1$ term should not change depending on how we code, but in the multivalue indicator case it will, because p-value is a function of the size of the effect, and the average differences between groups and a reference group will likely change dependent upon the reference group. For example, we have three groups, babies, teenagers, and adults, the average height difference between adults and teenagers will be smaller than between adults and babies, and so the p-value for the coefficient for the indicator of being an adult versus a teenager should be greater than an indicator of being an adult versus a baby.
Best Answer
Yes you always need to transform nominal categorical variables into dummy variable before including them in a regression model of any kind (including ordered logit).
The coefficients you get from any regression model always tell you how you expect the dependent variable to change (in some way or another) when you increase the associated variable "by one unit" (holding other variables constant). For a continuous variable like age or income a "one unit increase" makes sense - you get one year older or earn one more dollar. For a dummy variable like "female" or "recievedtreatment" that also makes sense - it means going from 0 (male/control) to 1 (female/treatment). But if your region variable is coded like 1=north 2=south 3=east 4=west then "increase by one unit" doesn't mean anything. Of course the model doesn't know that, so it will still give you coefficient telling you what will happen when you "increase region by 1" but the result will be garbage with no actual interpretation. So to deal with a variable like this you need to transform it into a set of dummy variables (leaving one out as the reference category).
And to reiterate - everything I just said is true of any kind of model: OLS, logit, ologit, poisson, tobit, whatever. In general, for most question about how you treat independent variables in a model (using dummy variables, calculating interactions, transformations, etc.) the actual model type is irrelevant. The same basic dynamics apply for all models (although the details of how they impact the DV will differ)