Should a colon be used after "for example"?

Either comma or colon is fine to use after the phrase "for example". Typically, a comma is used. When a more "significant" separation is desired, a colon may be used. Notably, if using a colon in your sample usage, the word that immediately follows should be capitalized since it starts a new sentence.

The research centre will use confidence intervals as well as numerical probabilities to to convey their findings. For example: Average surface temperatures will increase by two degrees by 2020 (95% confidence interval).

Colons are generally unnecessary. They sit in a liminal space between commas and periods. No one will miss them if you don't use them. Many books contain colons only on the copyright page. Yet the first book I grabbed to check doesn't have any colons on that page.

Semantically, your sample doesn't make sense because it's supposed to include "confidence intervals as well as numerical probabilities", but the "example" has neither. Confidence intervals would be expressed as ranges (eg, 1-3 degrees) or errors (eg, ± 1 degree). Probabilities would be expressed as percentages (eg, 75%), fractions (eg, 3/4), or ratios (eg, 3:1).

I would avoid using an example this way because it is meant to illustrate how data will be presented, yet appears like actual data. This potentially misleads readers into believing fictitious data.

The example sentence is also unclear. 2020 is about a month away. Is this 2-degree rise supposed to occur during the next month? Over this past year? Or some other period of time?