The Rune Scribe prestige class (from Unearthed Arcana: Prestige Classes and Rune Magic) has the 4th level Living Rune feature:

At the end of a long rest, you can choose to increase one ability score of your choice by 2 or increase two ability scores of your choice by 1. At the end of each subsequent long rest, you can alter this choice, reducing the scores you previously increased and improving different scores. (If you chose to increase two abilities, you can opt to alter only one of those choices.)

This feature is meant to be similar to the Ability Score Improvement feature that other regular classes have, but those features also specifically say:

As normal, you can't increase an ability score above 20 using this feature.

However, this ability from the Rune Scribe doesn't specifically say that you can't break the ability score limit of 20 - and I don't know how the prestige classes of previous editions behaved, hence my reason for the question.

Can the Living Rune feature increase your ability scores over 20?

No, it can't increase your ability scores over 20.

The normal ability score limit of 20 is reiterated in a few different places in the 5e rules. First, in Chapter 1 of the PHB/basic rules, under the "Class Features and Hit Dice" subheading below "Beyond 1st Level":

When your character gains a level, his or her class often grants additional features, as detailed in the class description. Some of these features allow you to increase your ability scores, either increasing two scores by 1 each or increasing one score by 2. You can’t increase an ability score above 20. In addition, every character's proficiency bonus increases at certain levels.

It reiterates that the normal limit for adventurers is 20 in Chapter 7, under the "Ability Scores and Modifiers" heading:

Each of a creature's abilities has a score, a number that defines the magnitude of that ability. An ability score is not just a measure of innate capabilities, but also encompasses a creature's training and competence in activities related to that ability.

A score of 10 or 11 is the normal human average, but adventurers and many monsters are a cut above average in most abilities. A score of 18 is the highest that a person usually reaches. Adventurers can have scores as high as 20, and monsters and divine beings can have scores as high as 30.

As shown by the above quotes, the text in the Ability Score Improvement feature that mentions the limit is actually redundant; even if it weren't in that feature's description, the ASI feature would still not let adventurers exceed the ability score maximum of 20.

Although Prestige Classes as described in that Unearthed Arcana document are not exactly normal classes, the UA describes them as following the multiclassing rules, with additional prerequisites. Nothing in the UA says their Living Rune feature lets them supersede the normal ability score maximum - so they must still obey that limit.

It wouldn't be unheard of for a class feature to increase a stat above 20.

Primal Champion page 49. PHB Barbarian level 20 feature.

Your strength and Constitution scores increase by 4. Your maximum for those scores is now 24.

However this is a level 20 feature and it specifically states that it increases the maximum of the score.

Page 173 of the PHB, under the Ability Scores and Modifiers section says: "A score of 18 is the highest a person usually reaches. Adventurers can have scores as high as 20, and monsters and divine beings can have scores as high as 30.

Considering specific beats general, since the Rune Scribe UA prestige class doesn't specifically say that it increases the ability score max, then it doesn't.