Ok, it took some time, it’s not as easy as it may seem.
I am not saying it’s the Best solution or only solution but hopefully it will be good enough and someone will use it 
We might implement a way better way to deal with it, in similar fashion like we have ‘no tweets’ message in twitter module, so for now we don’t have anything like this for ticker/dataset and again my solution is not perfect but it should do the work.
So example dataset:
Is it necessary to have separate column ‘NotBirthday’? No it’s not, but it clearer to see (at least I think so)
now for the filter:
(DAY(Birthday) = DAY(CURDATE()) AND MONTH(Birthday) = MONTH(CURDATE()) AND YEAR(Birthday) <= YEAR(CURDATE())) OR (DAY(NotBirthday) <> DAY(CURDATE()) AND MONTH(NotBirthday) <> MONTH(CURDATE()) AND YEAR(NotBirthday) > YEAR(CURDATE()))
So the message in ‘NotBirthday’ row will always be displayed, you absolutely can change it, here it’s just an example. (it’s not necessary to have YEAR there, but this way you can also enter actual birthday dates ie 1989-03-25 for example)
So for today 19.10.2015 assuming in appearance tab you have only ‘Name’ column.
it will display
guy a
guy b
No more Birthdays today!
for tomorrow, where none has birthday it will just display:
No more Birthdays today!
So you see it’s not perfect, but I believe it’s better than empty region. I had different ideas too, but it was too easy to prove that sooner or later it won’t work properly so the solution above works best and it’s not necessary to alter the data because of the filter.