Technically, “deontological ethics” is just a meta-ethical classification for ethical theories that evaluate the morality of an action according to a general rule/rules. (Though it is often used to refer to Kantian ethics, in particular.) If “follow these rules” is one of the rules of the theory, then following the rules would be a moral duty.
More generally, though, I think what you’re getting at is the “binding force” of normative claims. This would be shared by most (if not all) ethical theories that ascribe to an objective view of moral obligation (as well as some non-objective theories).
See the SEP subsection of the entry on Metaethics titled “Morals, motives, and reasons” for more information and further references.
To elaborate on how this would connect with one's attitude towards others, especially whether there are duties to enforce "the moral code", there are a number of interrelated issues. A few issues come to mind:
Prescriptive Universals: a view, sometimes called "moral universalism", is that the moral claim of realist theories are universal. This can mean several things: 1) they apply at all times, under all circumstances; 2) they apply to all relevant similar actions; and, most relevantly, 3) they apply to all people.
Tolerance: assuming that the moral rules are relevantly universal, the question of whether we have a duty to enforce them in the actions of others is closely tied to whether tolerance of conflicting views is obligatory, or even permissible. If it's not permissible to tolerate wrong action, then according to many views we would be obligated to (at least attempt) to prevent such action.
Moral Disagreement: there is disagreement over what is right/wrong, the pertinent question in this context is what the appropriate reaction to such disagreement is. Is it to "agree to disagree" (i.e., tolerate dissenting views), or can one side justifiably enforce their position?
- Contractualism: Some contractualists hold that we have a duty to prevent wrongs. Since violating the moral rules would be committing wrongs, we have a duty to prevent them. For example, there is Scanlon's Rescue Principle:
If we can prevent something very bad from happening to someone by making a slight or even moderate sacrifice, it would be wrong not to do so (Scanlon 1998, p. 224).
A stronger version is Ashford's Stringent Principle:
If we can prevent something very bad from happening to someone by making a great sacrifice (e.g., giving most of our income to aid agencies and spending a lot of our spare time on campaigning and fund-raising), it would be wrong not to do so (Ashford 2003, p. 287).
Similarly, in the entry on supererogation, the contract theorist Richards is described as giving the following account of supererogatory action (intuitively, an action that is "good to do" but which would be permissible not to do):
Thus, for instance, contract theorists (Richards 1971) describe principles of supererogation as those that ideal contractors in the original position would consent not to enforce in society. Supererogatory behavior is a requirement, but punishing those who do not fulfill the requirement would be too costly in terms of the relative pain incurred to the agent as against the benefit to the potential beneficiary. Failure to act supererogatorily is blameworthy and wrong, but lends itself only to informal criticism rather than to institutionalized sanction. Supererogation is justified only in qualified, circumstantial terms relating to the limited effectiveness of its enforcement. It has no intrinsic value.
This presumes a duty to enforce, but accepts that it has limits. The duty to enforce must be weighed against the cost of doing so.
While it's debatable whether this is a form of deontology (I could imagine definitions under which it wouldn't be) it is worth noting that it often takes inspiration from Kantian ethics, and that a similar "weighing problem" also arises in the Kantian setting. What are we to do when two duties conflict? Kant claimed that this didn't ever occur with "strict moral laws", but allows a wider sense of "duties of virtue" where such duties can conflict and our duties can be defeasible. In the case of a conflict between these "wider obligations", Kant discusses the "grounds of obligation" and says that the stronger ground prevails. Ranking the "grounds of obligation" according to strength requires something like a weighting of these various grounds. (See "Kantian Dilemmas? Moral Conflict in Kant's Ethical Theory" by Jens Timmerman; especially section 4, starting on page 51, for more discussion.)