Monks and politics

When I read through the Vinaya rules briefly I don't notice one which obviously prevents monks' being involved in secular politics.

DN 2 (see @Robin111's answer) suggests it's inappropriate, but I don't see a Vinaya rule on the subject.

Rājā Sutta (Ud 2.2) is similar:

Just now, lord, after the meal, on returning from our alms round, we were sitting gathered here at the assembly hall when this discussion arose: 'Friends, which of these two kings has greater wealth, greater possessions, the greater treasury, the larger realm, the greater stock of riding animals, the greater army, greater power, greater might: King Seniya Bimbisāra of Magadha or King Pasenadi of Kosala?' This was the discussion that had come to no conclusion when the Blessed One arrived."

"It isn't proper, monks, that sons of good families, on having gone forth out of faith from home to the homeless life, should talk on such a topic. When you have gathered you have two duties: either Dhamma-talk or noble silence."

“Is the Participation of Buddhist Monks in Thai Politics still Taboo?” by Katewadee Kulabkaew is on-topic:

Buddhist monks are known as world renouncers. In Thailand, the ideal monk has nothing to do with worldly matters, including impure activities like the lust for power or with materialistic concerns like politics. But throughout the past two decades, Thai monks are getting more involved in national politics and in the politics of Buddhism itself. The Thai public usually frowns upon clerics who interfere in such matters. However, political monks enjoy an increasing number of supporters.

The Thai constitution denies to monks the right to vote and thus strips them of the other political rights enjoyed by Thai citizens in general. The Thai state has decided to deny monks these rights with the ideological goal of protecting the purity of Buddhism – a pillar of the national identity and of the society’s moral consciousness. Some monks however have discovered a backdoor through which to play a role in politics.

Generally, monks’ political status depends on interactions between a Buddhist state and its sangha or monastic order, which vary across Buddhist societies and their distinct cultural and historical contexts. In Sri Lanka, Buddhist monks are allowed to run for political office and are free to establish their own political parties. As Sri Lankan monks have played an important role as nationalist leaders in many political struggles for Buddhist-Sinhalese dominance, there is public acceptance of their participation in politics. Sri Lankan Buddhists interpret the national sangha’s political activism as part of a holy and selfless mission, equivalent to practicing the Dhamma.

In contrast, such activities are considered unthinkable for Thai monks. Thailand has a centuries-long historical legacy of monastic orders controlled and disciplined by secular rulers. That legacy leaves no room for positive views toward monks that disobey the governance of Buddhist rulers, whose traditional mandate is to uphold and protect the religion.

Successive Thai governments have labelled monks’ political activism as improper and even something that could represent a security threat to the state. The Sangha Council has banned monks from participation in politics, issuing decrees to that effect at government instruction in 1974 and again in 1995. Last year the Supreme Patriarch signed yet another Sangha Council decree prohibiting political activities in monasteries.

So that seems to say that there a monks who want to be political, but who are not allowed to be, in Thailand.

Conversely there have been monks in Sri Lanka, associated with organisations which are described as political or nationalist -- for example some of these Sinhalese Buddhist nationalism organizations were founded by Theras.

There have also been some monks engaged in nationalist politics in Myanmar -- for example the "Ma Ba Tha" -- Wikipedia says that too (i.e. a bit like in Thailand) contravenes the organization which coordinates the Sangha with the State:

Some PAB members are connected to the 969 Movement. In May 2017, the State Sangha Maha Nayaka Committee, which regulates the Buddhist clergy, ordered the group disbanded. The group renamed itself as the Buddha Dhamma Charity Foundation, which government officials was also outlawed, according to government officials.

Therefore perhaps the Vinaya rule you're looking for (or the rule you'd expect) doesn't exist, and is instead a matter of interpretation (which varies), a topic for the Sangha to self-regulate, and/or to be regulated by the State.

Perhaps monks who are involved in "Buddhist nationalism" see that as (and if necessary justify that as being) dhamma-related and/or "dhamma-talk" (karaṇīyaṃ—dhammī vā kathā).

If there isn't a rule, possibly the topic didn't arise in the Buddha's lifetime. There are some stories of the Buddha meeting kings, but so far as I remember they tend to be like this one -- or the Buddha once tried unsuccessfully to stop a war (maybe that counts as being involved with politics).

In DN2, the Samaññaphala Sutta: The Fruits of the Contemplative Life, the Buddha speaks of virtues in three sections of his discourse; The Lesser Section on Virtue, the Intermediate Section on Virtue, and The Great Section on Virtue. Part of the Intermediate Section on Virtue reads:

"Whereas some brahmans and contemplatives, living off food given in faith, are addicted to talking about lowly topics such as these — talking about kings, robbers, ministers of state; armies, alarms, and battles; food and drink; clothing, furniture, garlands, and scents; relatives; vehicles; villages, towns, cities, the countryside; women and heroes; the gossip of the street and the well; tales of the dead; tales of diversity [philosophical discussions of the past and future], the creation of the world and of the sea, and talk of whether things exist or not — he abstains from talking about lowly topics such as these. This, too, is part of his virtue.

If even speaking of kings, ministers of state, armies, and battles is considered lowly, then it seems implicit that to be involved in politics and governments would also be lowly and not virtuous for one "living off food given in faith" i.e. an almsman or monk.