I have just answered your support ticket on a similar matter and I believe the following part from it is relevant here in case somebody else stumbles upon a similar problem:
What I believe is happening in your case is that the button of the calendar has its own tooltip (i.e. the HTML element has the title property set) which is picked up by the tooltipmanager that is configured to transform all regular tooltips on the page to RadToolTips. This happens because the RadToolTipManager tracks AJAX requests and also tooltipifies content received via AJAX.
You can only have one RadToolTip visible on the page in the same way you can have only one browser tooltip visible, so when you hover the mouse over the button a new tooltip (from autotooltipify) attempts to open.To do so it closes the previous one, which is perhaps the behavior you are observing.
Currently this scenario is difficult to achieve and I suggest we continue the discussion in the support ticket. In the meantime you can try setting the ShowEvent of the automatic manager to OnClick so that it will not hide the calendar until one of its items is clicked, which may be desired at some point.
Generally, for the time being I would suggest that you remove the manager with AutoToolTipify functionality and if possible use target controls for it. You can set the needed tooltip string as the value each target control holds and use the OnClientShow event to set it as text (so that you avoid AJAX requests just to get a simple string):
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