Thank you Sun Lion for the article, which primarily mentioned about Harry's relationship with Prince William and to some extent Prince Charles.
I do think Harry's relationship with Princess Beatrice and Princess Eugenie might also be strained. The main reason is that Meghan allegedly (according to the Sun) drop their names in a court case to prove that some members of the Royal family are able to work independently whilst holding a title. Beatrice and Eugenie of course were never working members of the royal family and don't use their title for businesses/advertisement. The other reason (which could be speculative) is that in Finding Freedom, Omid Scobie and Caroline Durand supposedly mentioned that Harry and Meghan were apparently very angry that The Duke of York, Princess Beatrice and Princess Eugenie leaked their relationship. Harry and Meghan have denied their involvement with Finding Freedom, but to have your names dragged in the book (similar like way The Duke and Duchess of Cambridge have been portrayed), must have angered the York Princesses, especially if they did not leak the relationship themselves.
I think the Sussexes' relationship with the Wessex family, Princess Anne's family (including the Phillips and Tindalls) may not be positive either. At one point, Meghan's court case apparently (again, according to The Sun) mentioned that the Royal Family (Not sure if it's as an institution or just family members?) have been unsupportive of them. Mike Tindall and Sir Timothy Laurence at some point have mentioned that the Royal Family members have been very welcoming. I do think the Wessex family, Princess Anne's family (Phillips and Tindalls) might be upset by those accusations, given that they have tried their best to include Meghan into the fold. Of course, Meghan may have been accusing the courtiers and palace staffs.
At this point, I think it's not just tension between Harry & Meghan and William & Catherine, but rather Harry & Meghan and other members of the Royal Family (though I am not sure about the relationship with the Snowdons, Chattos, Gloucesters and Kents)