Do foreigners need to change their name to a Japanese name when they become Japanese citizens?

You can have any name you want, and it can be as long as you want, providing four conditions:

1. 

   None of these characters are allowed
   in a Japanese 戸籍 {koseki} (family register).

   It has to be written exclusively in or a combination of (modern, not archaic) hiragana, katakana, or kanji. Just like you can’t write a name on American official documents in Thai, Cyrillic, Hangul, or Arabic … or even use diacritic marks used in European languages (ex. the German sharp S [“ß”] or French cedilla [ç] or others [ÂÄÅÀÁ]) — people becoming American must adapt their name to the plain 26 character alphabet — you have to adapt your name to Japan’s writing system.
   
   Because there is not a perfect one-to-one correspondence between either Japanese language "spelling" or pronunciation and other languages, most non-Japanese names will need to be adapted or modified for the new writing system. "Adrian" for example would become 「エイドリアン」 which when literally converted to the Latin / "Roman" alphabet (ローマ字 {rōmaji}) would be written as "Eidorian".
   
   If you're lucky, there are some names that can convert to Japanese and back with little Lost In Translation. This is the exception, not the norm.
2. Like a newborn child, you will be strongly discouraged from choosing a new name which will cause problems in society. For example, intentionally choosing the name of a unique famous person for the purpose of fraud / identity theft, or choosing a provocative name like “demon” (悪魔 {Akuma}) or “dickhead” or “emperor” is discouraged.

3. 

   Most Icelanders do not have surnames.

   You get one family name, and one given name. No middle names, additional given names, or mononyms (single names, like “Madonna” or “Cher” or “Prince” or “Björk” †). If you want additional names, they are concatenated all together in the “given name”: no spaces, dashes, punctuation, or other forms of separators are allowed. For example, the given names on a British birth certificate “Adrian Charles George” would become the one name 『エイドリアンチャールズジョージ』“Eidorianchāruzujōji” in 仮名 {kana} (Japanese syllabet).
4. If you are legally married in Japan and/or with children and these people are Japanese nationals, you must all have the same family name. That means either the man takes the woman’s name, the woman takes the man’s name, or you choose a brand new family name for the family register and everybody takes that name.

Note the name in parentheses.

Note that this is for official domestic-use identification. For identification that is used for overseas outside-of-Japan purposes (ex. a passport), it is possible to have a Latin (26 letter alphabet) letter name, alternate names, and non-Japanese spellings. Example: your legal domestic Japanese name may read 『佐藤キンバリー』“Satō, Kimbarī” (in 仮名／漢字kana/kanji), but your Japanese passport could say “SATO(SMITH/PARKER), KIMBERLY(KIM)” if you can prove you use/need those foreign names on your for-foreign-use passport.

† Legally speaking, these celebrities actually do have more than one name on their official documentation, even though in public they are only known by one name.