Do paladin oaths have to be taken seriously?

Paladin Oaths are meant to be taken seriously

The choice of oath is a significant part of the paladin class and it is meant to be a serious choice. The PHB suggests that even before reaching third level and choosing the oath as a class feature, you should probably have read ahead and thought about which choice to take depending on the nature of your paladin. Furthermore, the sidebar BREAKING YOUR OATH states that:

A paladin tries to hold to the highest standards of conduct, but even the most virtuous paladin is fallible. [...]

A paladin who has broken a vow typically seeks absolution from a cleric who shares his or her faith or from another paladin of the same order. [...]

If a paladin willfully violates his or her oath and shows no sign of repentance, the consequences can be more serious. At the DM’s discretion, an impenitent paladin might be forced to abandon this class and adopt another, or perhaps to take the Oathbreaker paladin option that appears in the Dungeon Master’s Guide.

If your friend's paladin chooses an oath but disregards the oath's tenets and has no intention of adhering to them, I think that would be fairly described as wilfully, unrepetantly violating the oath, and so that paladin's continued paladinhood is called into question - the DM can force them to change class based on their behaviour.

This is a quite watered-down version of rules from prior editions - 3e strictly stated that a Paladin who transgresses against the class's vows loses their paladin abilities and must atone in order to regain their powers. 5e is less strict about this, placing the matter at the DM's discretion; however, it is still clearly intended that the Paladin is meant to stick to the vows of their oath.

If your DM and your table don't care about the tenets of the various oaths, that's entirely your business. If your group doesn't find worrying about paladin oaths to be any fun and you'd rather just keep the mechanics and abilities but play differently, that's fine, and that's a valid way to play.

However, the default assumption is that oaths are important, and people on the internet who don't know your group can't tell you whether your friends think differently! If I were a paladin player taking my oath seriously, I would be very annoyed if other paladins got to break their vows with impunity, and if I were a player ignoring my oath I would not enjoy the DM unexpectedly telling me that I needed to change my character.

Ultimately, the discretion lies with the DM, and your friend needs to talk to them about whether or not they can play this way. Ideally they need to make sure that everyone playing is on the same page about this, especially if they're all playing paladins.

A paladin must adhere to their oath, although they don't specifically need to be lawful good.

Before it was edited, this question asked if the paladin has to be lawful good. It was later clarified that they want to take Oath of Devotion, whose ideals suggest lawful goodness. It's worth noting that you don't actually have to be lawful good to be a paladin in D&D 5th edition, and technically you don't actually need it to take Oath of Devotion.

A paladin in D&D 5th edition takes the Sacred Oath class feature when they reach 3rd level. The fact that it's called sacred oath strongly suggests that it's meant to be taken seriously.

The penalty for disobeying an oath, and determining when you have disobeyed it, are largely up to the DM. According to Player's Handbook p.86, sidebar "Breaking Your Oath", you might lose your paladin status if you wilfully violate your oath and show no sign of repentance. (Carcer's answer covers this scenario very well, and deserves your upvotes.)

Oath of Devotion requires you to adhere to the tenets of honesty, courage, compassion, honor, and duty. Breaking any one of those directly (i.e. lying, being cowardly, failing to show compassion, acting dishoborably, and shirking duty) is a violation of your oath. You don't strictly have to be lawful good—perhaps your character focuses on some of the tenets over the others—but it's right there in the rules that you have to adhere to the tenets of the oath you swear to.

If you don't want to follow this oath, there are others in the Player's Handbook, in the Unearthed Arcana documents, and in other sourcebooks. The core rules include:

• Oath of the Ancients: This is explicitly good (it refers to mercy, kindness, and forgiveness), but not lawful. In fact, chaotic good would work well for this.
• Oath of Vengeance: These are called dark knights, and don't even sound necessarily good aligned. They destroy their enemy by any means necessary. It sounds like they can even commit evil acts as long as it's in the greater good.
• Oathbreaker: An explicitly evil paladin in the Dungeon Master's Guide. Although you don't automatically become an Oathbreaker when you break your oath, it does note that a paladin can fall from grace, and that you can become an Oathbreaker if you break your oath in order to deliberately serve evil. It's also unlikely for an Oathbreaker to work well in a party with other lawful good paladins.

Unearthed Arcana also has the Oath of Heroism (now Oath of Glory in Mythic Odysseys of Theros), Oath of Treachery, Oath of the Watchers, Oath of Conquest (updated in Xanathar's Guide to Everything) and Oath of Redemption (also updated in Xanathar's). Further sourcebooks may provide other options, but a complete list of all Paladin oaths is beyond the scope of this question.