Tests to distinguish between PRNG and a CSPRNG What are the tests which a regular PRNG would fail but a CSPRNG would succeed? Is it just the next-bit test or are there multiple other tests which a PRNG which is not a CSPRNG would fail? Would a PRNG which is not a CSPRNG pass the Chi Square tests?
What are the tests which a regular PRNG would fail but a CSPRNG would succeed?

The goal of a PRNG, and the duty of a CSPRNG, is to have an output that can't be distinguished from true randomness (with a significant advantage, by a program that can actually be run). Thus it's a failure of a PRNG, and a capital offense for a CSPRNG, if it does not pass all randomness tests it receives (assumed correct and well used, including correctly interpreting the results, which by nature are uncertain to a degree).
Nevertheless, a few bad PRNGs which fail some generic statistical tests like Ent, Diehard, Dieharder, TestU01, NIST SP800-22 rev 1a. More PRNGs fail only specific tests designed with knowledge of the PRNG design.
A well-designed and well-implemented CSPRNG will, by definition, pass all tests. That's including the above list of tests. We can have a (faint) hope that running some tests in this list disprove that a PRNG is a CSPRNG.
Caveat: that's essentially useless to argue (much less, prove) that a PRNG is a CSPRNG (which requires analysis at the design) or well implemented (which require looking at the implementation, and test vectors). Much like analyzing statistics on winning a lottery is not useful to argue that this lottery system is run honestly.
Addition (as rightly suggested by @SEJPM): We can't stress it too much, don't count on generic statistical tests to find weaknesses in PRNGs, or worse CSPRNGs (rather, count on CSPRNGs to help debug statistical tests). A CSPRNGs requires a sound design, and years of analysis, of its design and/or of the primitive(s) it uses, by smart motivated cryptanalysts. Only after they failed to find a theoretical distingisher do we have a serious candidate CSPRNG.


Would a PRNG which is not a CSPRNG pass the Chi Square tests?

That depends on PRNG and test parameters (that might also depend on the seed used). For many non-crypto PRNGs in actual use, and all CSPRNGs, yes. For others, that depends a lot on if the test is run for a long enough sample, and on if the χ2 test is bidirectional or not.


Why do you need to know the PRNG Design?

In cryptography, adversaries are assumed to know everything about the system attacked except the key (that is, for a PRNG, the seed), by Kerckhoffs's principle. They will design clever algorithms, perhaps dedicated hardware (e.g. the bombe), in order to break the system (that is, for a PRNG, distinguish the output from random, or worse predict future output from past output).
And in practice, that makes a helluva of a difference. It's easy to devise a test that predicts the next outputs of MT19937 knowing earlier ≈19968 bits, using a specifically designed test for that generator. Most tests not specifically designed for the class of generators MT19937 belongs to will give its output a clean bill of health.

Why can't the next-bit test be used to distinguish a CSPRNG from a PRNG without knowledge of the PRNG design?

The next bit test is theoretical. Unlike the 5 families of statistical tests I linked to, it's not a program that can be run to completion on a sample of the output(s) of the PRNG tested.
No practical test on the output of a candidate CSPRNG can confirm it's one.