Will removing a column having same values for all observations affect my model?
If one of your column in the dataset is having the same values, you can drop this column as it will not do any help to your model to differentiate between two different labels while on the other hand, it can even negatively affect your model by creating a bias in the data.
For Example: Consider you have two different fruits, like one is Green Apple and one is Guava. Then, both of these fruits will have the same color i.e. "Green", so that basically means that you just can not differentiate both these fruits on the basis of their color, but if they have been two different colored fruits, you could have used this feature to differentiate between them.
Hope it helps clarifying what you should do with such a column with same set of observations.
Thanks.
One of the columns in my dataset has the same value for all observations/rows. 
Should I remove that column while building a machine learning model?
Will removing this column affect my model/performance metric?
If I replace all the values with a different constant value, will it change the model/performance metric?