If someone offers his hand for shaking on Eid Day for example, what should one do?

Answer

It will be permissible, in fact necessary, to politely 'rebuff' shaking hands where it is bid'ah. If one shakes hands because of one's own weakness, it will not be permissible to justify the wrong act with fanciful interpretations. However, if the person is a musaafir or one whom you have not met for a long time, then it will be permissible to shake hands since in this instance it will not be fulfilling a customary bid'ah practice.