If apex:actionSupport is not working for apex:inputField, use apex:actionRegion.
Sample Code:
<apex:actionRegion>
<apex:inputField value=”{!opportunity.stageName}” id=”pb”>
<apex:actionSupport event=”onBlur” rerender=”table” status=”status”/>
</apex:inputField>
</apex:actionRegion>
There is problem when using required = true and Immediate = true while using action support,it can be a reason why your action support is not working. you need to use Action region to achieve the functionality.