What does 'passing away of dhamma' mean in Satipatthana sutta?

"Passing away" is the Pali "vaya", as found in the Buddha's last words, as follows:

‘Conditions fall apart. Persist with diligence.’

vayadhammā saṅkhārā, appamādena sampādetha.

DN 16

It appears by the content of the Satipatthana Sutta the term "Dhamma" refers to various "teachings", "principles" or "truths" of the Buddha, such as the seven factors of enlightenment and the four noble truths.

Thus, monks such as Bhikkhus Buddhadasa & Sujato translate Dhamma here as Truths or Principles. However, is should noted the common illogical translation here of "dhamma" is "mental objects".

Regardless of the translations, since the development fruition of the seven factors of enlightenment and the realisation of the four noble truths are permanent in the mind of an Arahant, obviously these dhammas cannot be subject to passing away/destruction (vaya).

For example, MN 12 says about an Arahant's realisation:

Sariputta, even if you have to carry me about on a bed, still there will be no change in the lucidity of the Tathagata's wisdom.

SN 12.20 (about Dependent Origination) and AN 3.136 (about the Three Characteristics) say about the Dhammas an Arahant realises:

Whether Realized Ones arise or not, this law of nature persists, this regularity of natural principles, this invariance of natural principles.

Also, the term "arising" ("samudaya") is explained in SN 22.5 as when a monk "approves, welcomes and keeps clinging" to the aggregates. It follows, for one that has entered the Path, clinging to the seven factors of enlightenment and the realisation of the four noble truths is as equally ridiculous as these dhammas passing away.

These obvious points shows how questionable the Satipatthana Sutta is. Bhikkhu Sujato called the Satipatthana Sutta the "Piltdown Sutta". "Piltdown" was a term used in a certain scientific fraud. Yet my answer will probably receive lots of down votes by those who worship ink & paper and who worship the Burmese gurus who worship the (fraudulent) Satipatthana Suttas.

The word "dhamma" has many meanings in the Pali Canon, as seen in this SuttaCentral dictionary entry.

The "dhamma" you are referring to from the Satipatthana Sutta is most likely referring to:

• a constituent of experience; an aspect or quality of existence; physical sensation; a mental state or quality (good or bad); (sometimes merely) thing, phenomenon, matter; the nonindependent, conditioned constituents of processes and events, progressively more and more minutely analyzed into fundamental types of event or fundamental regularities.

• mental constructs, concepts, ideas, what is to be cognized by the mind, that which is the object of mental activity.

From Ven. Yuttadhammo's book How To Meditate, chapter 1:

The fourth foundation, the “dhammas”, contains many groupings of mental and physical phenomena. Some of them could be included in the first three foundations, but they are better discussed in their respective groups for ease of acknowledgement. The first group of dhammas is the five hindrances to mental clarity. These are the states that obstruct one’s practise: desire, aversion, laziness, distraction, and doubt. They are not only hindrances to attaining clarity of mind, they are also the cause for all suffering and stress in our lives. It is in our best interests to work intently to understand and discard them from our minds, as this is, after all, the true purpose of meditation

The materialistic mindset invented language built around the key notion of objects. Objects are represented by nouns.

In contrast to that, the spiritual mindset or the mind-over-matter mindset came up with the concept of dharma as its primary building block. Dharmas are not nouns because they are not objects. Instead, dharmas are adjectives because they represent qualities.

In the mind-first view of existence, the primary building block is an experienced quality, a dharma.

Dharmas arise into existence, stay for some time, and then disappear. Most dharmas don't arise momentarily but gradually form from some combination of previous dharmas. Then they mutate or change, often weakening over time. Eventually they vane away or gradually decompose and become components for other dharmas.

All dharmas, practically speaking, with some rather theoretical exceptions, are transient. This is because, being nothing but qualities of combinations of other qualities they morph away out of existence as their underlying qualities morph away as their underlying qualities morph away and so on.

When Buddha describes meditation on arising and passing away of dharmas, he is referring to gradual formation, deformation, and eventual morphing out of existence of all qualities observable by the meditator.

This includes so-called external qualities, and so-called internal qualities. "External" qualities include shape, size, color, loudness, pitch, smell, temperature etc. - morphing in and out of existence either quickly or slowly. "Internal" qualities include discursive, emotional, somatic, conceptual, abstract, intuitive etc. characteristics, morphing in and out of existence. Qualities in both groups arise and vane due to forces out of our control (such as the time passing) as well as due to forces in our control (such as us moving our gaze or our attention around).

This is what's called "arising and passing away of dhammas".

Specifically in context of Satipatthana the external dhammas are analyzed by their sensory basis (visible, audible, tactile etc) and the internal dhammas are analyzed by their effect on one's practice. The most important dhammas to watch are the qualities of mind that help or harm meditation, such as sensuality, negativity, torpor, anxiety, uncertainty, as well as the positive qualities such as mindfulness, determination, enthusiasm, focus, calmness etc.

The most abstract method of contemplating the qualities is by observing the Noble Truths in action, happening in front of one's very eyes. "This is the quality of dukkha (i.e. ~painful aspect of some experience) arising, this is the underlying quality of craving, here the quality of craving has been abandoned, here the dukkha quality is no longer to be found."