Strawberry Harvest in the Eighth Year

Answer

According to the Torah, the shemitah-status of vegetables follows their time of harvest: if harvested during shemitah, they have kedushah. In contrast, if harvested during the eighth year, they would not have kedushah and are obligated in terumot and ma'aserot with a blessing.

There is one exception to this rule: a vegetable that has completed its growth at the end of shemitah, but harvested only in the eighth year (I didn't want it to spoil over Rosh Hashanah, for commercial reasons, etc.). For such vegetables (relevant also to bananas), separate terumot and ma'aserot without a berachah, but handle it as having kedushat shevi'it miderabanan, since it completed its growth during shemitah.
In The Consumer's Guide for Shemitah, chapter 27 B1, we write that this applies if the majority of the vegetable's growth occurred during the shemitah year, but this is a stringency. The baseline halachah (ikar hadin) is that only if the vegetable completed its growth during shemitah would it still have kedushat shevi'it.

This applies only to vegetables that are not subject to the sefichin prohibition (that were not planted during shemitah). Vegetables that were planted during shemitah in a forbidden manner are still forbidden for consumption even if harvested during the eighth year (ibid., quoting the Chazon Ish 9:12). Some are lenient and hold that after Chanukah it is permissible to eat them. See the dispute whether sefichin applies only to vegetables that completed their growth during shemitah; vegetables that most of their growth occurred during shemitah; or even those that just began to grow during shemitah. We are stringent, following the last listed opinion).

Bottom line: The first week after Rosh Hashanah, separate terumot and ma'aserot without a berachah and treat the produce as having kedushat shevi'it. For harvest 8 Tishrei and on, the produce no longer has kedushat shevi'it and terumot and ma'aserot should be separated with a berachah.