The EXIF data in a picture contains multiple sections. Adobe (and most other software makers) copy only the standard generic picture info from the original file into the exported file. Sections like the "maker notes" are not copied because Adobe does not know what the data is. (Not that they need to understand it; just copy it....)
To build on ManiacJoe's reply, Time Zone and Ambient Temperature are both non-industry-standard tags, stored in the MakerNotes section for manufacturer-specific proprietary tags. To copy one or more such fields from the master file to the exported copy, you could use the Run Any Command plugin as a post-process action in your export; you'd configured it to run the free Exiftool utility to copy the fields.
Ugh. Neither of these are correct answers.
The exif standard was updated in July 2016 and includes both of these tags. It includes Time Zone Offset (exif tag: 0x882a), Ambient Temperature (exif tag: 0x9400) among others. You can learn about exif tags here or here. I'm not referring to Makernotes, xmp or iptc. Just exif.
I use Exiftool to copy the standard from my raws to my exported jpegs, but it would be better if Lightroom supported the standard. I don't use the plugin, just the command-line tool.
It would also be helpful if people here would give correct information.
Thanks, I had missed the update to Exif 2.31 (http://www.cipa.jp/std/documents/e/DC-008-Translation-2016-E.pdf).
Which camera(s) are having their Exif 2.31 fields stripped? I downloaded sample raws from dpreview.com for some new cameras (Canon EOS M5, Pansonic Lumix DMC-FZ2500, Olympus OM-D E-M1 Mark II, Nikon D3400), and only the Sony a6500 used Exif 2.31.
If you really want to copy a tag named "Time Zone", then you are probably seeing a proprietary TimeZone MakerNote tag, e.g. from Canon or Nikon. As mentioned above, LR doesn't export MakerNote tags. Exif 2.31 doesn't define a TimeZone tag, but it does define three tags that provide the time-zone offsets: OffsetTime, OffsetTimeOriginal, and OffsetTimeDigitized. But you mentioned "Time Zone", not those tags.
AmbientTemperature is also a proprietary MakerNote tag for Sony and Reconyx. I downloaded a sample file from the new Sony a6500, and even though the file is Exif Version 0231, AmbientTemperature is only stored in the MakerNotes. Which camera are you using that stores AmbientTemperature in EXIF:AmbientTemperature?
I did observe that LR doesn't export EXIF:AmbientTemperature and the EXIF:OffsetTime tags. You can file a feature request here for LR to support the new EXIF 2.31 tags: Photoshop Lightroom | Photoshop Family Customer Community .
Finally, you might catch more flies with honey than vinegar.
The exif standard was updated in July 2016 and includes both of these tags. It includes Time Zone Offset (exif tag: 0x882a)
EXIF:TimeZoneOffset is not defined in EXIF 2.3 or 2.31. It is a non-standard tag that some manufacturers used to record time zone. It wasn't very well thought-out, since it only allows for offsets in units of hours, whereas over 1 billion people live in time zones with fractional hour offsets.