1. Pronunciation of the 'z'
It is probably in emulation of Northern/Central Spanish that the British pronunciation is as such (though, tellingly, the initial 'I' is pronounced as the diphthong /aɪ/
), as indeed the OED claims in its entry:
In British English, the name is usually pronounced in an approximation of the Spanish.
2. Older toponyms
Ibiza wasn't always known as such in English. In the 18th and 19th centuries the island was known to the British, and especially to the Royal Navy, as Ivica (from the older Spanish orthography Iviça):

Looking at Google Ngrams for the three toponyms, we see that Iviza became the more popular name during the end of the 19th century, and Ibiza overtaking the older words in the 1930s-40s:

This adoption of the current Spanish orthography was possibly accompanied by the 'Spanish' pronunciation, the word only relatively recently gaining popularity.
Etymology
The toponym has gone through a number of modifications both before (as different peoples had control or influence on the island) and after entering the English and Spanish lexicons:

- The change 'b' > 'v' in Catalan was due to hypercorrection in an attempt to 'de-arabic' -ise words.
Different Pronunciations
Note, the native language of Ibiza is Catalan, not Spanish, and in Catalan the island is called Eivissa.
With the diphthonged initial 'I' and the distinción pronunciation of 'z', the British pronunciation of the word ends up sounding like a hybrid of the Catalan and Spanish words:
- Catalan: Eivissa
[əjˈvisə]
[əjˈβi.sə]
- Spanish: Ibiza
[iˈβiθa]
(Northern/Central Spain)
- Br. Eng.: Ibiza
/aɪˈbiːθə/
- Am. Eng.: Ibiza
/ɪˈbiːzə/
/iːˈbiːsə/
3. Other loanwords
'Z' is, to my knowledge, not pronounced as /θ/
in any other word in English.
However, the attempt to evoke the native pronunciation of a loanword happens occasionally in other toponyms. The word Weimar, for example, entered the English lexicon around the same time as Iviza/Ibiza, and it is commonly pronounced in British English as /ˈvaɪmɑːɹ/
, emulating the German pronunciation of 'w'.