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In the United States, accepting a pardon is an admission to the crime. In Burdick_v._United_States, the majority opinion stated that a pardon "carries an imputation of guilt; acceptance a confession of it."

The United States has dual sovereignty. Each state and the federal government makes and prosecutes its own laws. A federal pardon does not grant immunity to state laws, and vice versa. A person pardoned in one jurisdiction can still be prosecuted for the same crime (e.g. tax evasion) in another jurisdiction.

Could the acceptance of the pardon then have any bearing on the case in the other jurisdiction? Can the admission of guilt be used in the new case? Can it be used as "reasonable cause" for various actions? Does the defendant lose their right to refuse to testify in the new case?

Could the acceptance of the pardon then have any bearing on the case
in the other jurisdiction?

Possibly, but not much. There is very, very little case law on this point since: (1) pardons are rare (especially federal ones), (2) people who are pardoned generally do so because everyone in the criminal justice process in the prior case agrees that the person is reformed and they are usually correct, (3) the statute of limitations has often run on a new prosecution, and (4) many cases where these issues arise, are probably not appealed (either because the neither parties attempts to, or because a defendant is acquitted and there is no appeal), but an appeal is necessary to give rise to binding precedents.

Can the admission of guilt be used in the new case?

The criminal collateral estoppel effects of a pardon flow from the adjudication on the merits which is vacated. Also, comity between sovereigns and public policy mitigate such a rule. The pardon power would not be very interesting if it routinely resulted in a new prosecution that was conducted on a summary basis via a preclusion doctrine such as collateral estoppel.

In many cases, the statute of limitations will have run on the original crime or there will be no parallel state law crime, but this is not always the case.

Also, I would disagree with the statement that a pardon always implies an admission of guilt to the crime for which a pardon was granted, even though that statement is often used rhetorically.

For example, one important use of the pardon power is to commute the sentence of someone who asserts that they are factually innocent but have been convicted of a crime, potentially in a manner that is not subject to further judicial review, and treating that as an admission of guilt doesn't make sense.

As the Wikipedia entry on Burdick notes in the pertinent part:

Legal scholars have questioned whether that portion of Burdick
[ed. about admission of guilt] is meaningful or merely dicta.
President Ford made reference to the Burdick decision in his
post-pardon written statement furnished to the Judiciary Committee of
the United States House of Representatives on October 17, 1974.
However, said reference related only to the portion of Burdick that
supported the proposition that the Constitution does not limit the
pardon power to cases of convicted offenders or even indicted
offenders.

I would read this a dicta, as this portion of the holding was not necessary for the court to reach its conclusion and the fact pattern is an usual one that doesn't raise the pertinent concerns when a pardon is requested following a conviction based upon a claim of innocence. The holding of Burdick was that there was no pardon because the pardon was rejected (in a manner very similar to a common law disclaimer of a gift), so it didn't reach the effect of a pardon that is accepted.

Can it be used as "reasonable cause" for various actions?

This is a bit too vague to know what you are getting at.

I suppose that a pardon could constitute reasonable cause for some things favorable to a defendant who is pardoned (e.g. potentially in a motion seeking to reopen a termination of parental rights entered on the basis of the conviction).

I suppose it could also be used in a manner potentially unfavorable to a defendant (e.g. showing a pattern of past conduct that demonstrates modus operandi in connection with a prosecution for a new crime). I don't think it could be used as grounds to deny an occupational or business license for bad character.

Still, without more clear context it is harder to know what you are really looking for in this regard and I'm not confident that my examples address that.

Does the defendant lose their right to refuse to testify in the new
case?

I haven't reviewed the case law, but my intuition is that if it has never been waived before, it wouldn't be waived by the pardon, but that if it was waived in a previous proceeding resulting in a conviction that was then pardoned, that the prior sworn testimony might be admissible evidence in the new action since it is not hearsay and isn't itself evidence of a prior conviction. The context of the prior testimony might have to be concealed from the jury.

As noted by @Putvi, the defendant could not claim risk of conviction for the federal crime as a ground for invoking the 5th Amendment if a pardon is accepted (something that is implied in Burdick), but if there was an overlapping state law crime, risk of conviction for the state crime could constitute a grounds upon which to invoke the 5th Amendment.

Burdick does stand for the proposition that a pardon not solicited by the defendant, that is rejected, cannot provide a basis for removing the 5th Amendment protection with respect to a risk of conviction for federal crime.

I would also be inclined to think that matters disclosed in an application for a pardon might be admissible evidence as a non-hearsay statement of a party-opponent, if the statement was stripped of the pardon application context (which would be unduly prejudicial since it would imply a prior conviction which otherwise wouldn't be admissible).

The reason a person can be compelled to testify after receiving a pardon is that they are no longer in jeopardy of incriminating themselves. http://time.com/4868418/donald-trump-presidential-pardons-backfire/

It would depend on the specific situation, but if you were in jeopardy of incriminating yourself in the state trial, you could plead the fifth, just like the Time.com article's hypothetical about testifying before congress.

You could bring up the pardon in the state court, if the judge allowed it, but that doesn't mean you are guilty just because you accept a pardon.

But Burdick was about a different issue: the ability to turn down a pardon. The language about imputing and confessing guilt was just an aside — what lawyers call dicta. The court meant that, as a practical matter, because pardons make people look guilty, a recipient might not want to accept one. But pardons have no formal, legal effect of declaring guilt.https://www.chicagotribune.com/news/opinion/commentary/ct-perspec-pardons-presidential-trump-nixon-ford-kardashian-0608-story.html

Whether it made a difference would be up to the jury, in that they would have to decide if it was relevant evidence that aided their deliberations.