For the logic chips, there may be some "testers", though there main purpose would be more like EPROM programming. So it more like a universal programmer for EEPROMs and the like, that can also test standard (e.g. TTL, 3 / 5 V CMOS) logic.
For most other chips I am with Brumby - build a test board for that specific chip.
For OPs there might be a certain sense to have one around - as they are common and relatively easy to check.
For an OP I would use something like a x 100 amplifier from a switchable impedance ground. So one could test thinks like offset, bias and maybe noise, besides the basic go / no go test. It would also need a suitable supply choice (e.g. 5 V or some maybe 12 V, as some OPs are 5.5 V max. and others could be 8-10 V minimum).