First, the answer to this question depends to some degree on the type of IRT model. The Rasch measurement model holds that item calibrations will be invariant across different subgroups (schools) even when these groups differ regarding their level on the measured trait. So, if the only difference between students in schools A, B, and C are the average level of math ability, then there would be no need to adjust item calibrations. The possible exception to this is when, because of the nature of the sample, some items may not be properly calibrated due to floor or ceiling effects (items that few if any students in the sample answer correctly or nearly every student answers correctly). Other IRT models (e.g., 2- and 3-parameter models) do not have the sample-free properties of the Rasch model. Second, the salient question in your example must do with the presence of differential item functioning related to different levels of exposure to computerized adaptive testing. You would need to assess for DIF and either remove any DIF items or have multiple calibrations per item based on the student's experience with CAT. My guess would be that exposure to CAT would primarily affect the first few items to be administered. In fact, you might want to consider providing students with a practice test to help them become familiarized with CAT.

@Barth, thank you so much for your response. Suppose I am using the Rasch model to calibrate the items then the results will be sample invariant ( this is is the assumption(. Don't I need to validate these assumptions?

In addition to that suppose it turns out that two samples giving two different parameters for the same item (under Rasch model). In this case use DIF? 

Yes, the Rasch model is sample-free and test-free provided that (a) the data fit the Rasch model (as indicated by item and model fit) and (b) there is no DIF present in the items. So assessing DIF is important to determining if the measure has the property of being sample free.

Personally, instead of finding the DIF of the item. You may try IRT equating scale scores as well as the parameters by eliminating the difficulty effect. Otherwise, it is impossible to compare different different groups. On the other hand, as for the CAT, since your have got your calibrated item bank. I estimated that you what you employed was a common-item randominzation design. In that case, please equate all the items, scale cores and IRT parameters before comparison. In such a scenario, you are NOT looking into the differences. Rather, you are looking into the commonalities as well as comparisons. Therefore, the equating mindset could be employed and would be very different from DIF.