Revenge is profitable, gratitude is expensive.

This quote refers to civil wars and the tendency after them to remember the injuries caused so that these can be revenged but not what the victors owe to those whose actions helped them. Injuries are remembered because revenge justifies taking spoils from others. The need for gratitude is forgotten, because then you might be expected to give a gift, which could cost you. In other words, we nurse the emotions that lead to our material betterment.

The ascent to greatness, however steep and dangerous, may entertain an active spirit with the consciousness and exercise of its own power: but the possession of a throne could never yet afford a lasting satisfaction to an ambitious mind.

This is the sad or "melancholy" plight of the Emperor Severus, who cannot be satisfied with what he has even in old age. When old and infirm, he desires heirs, so he marries and has two sons, who turn out to be spoiled and entitled creatures and a disappointment to him. They also hate each other. This causes division in the court and troubles that could have been avoided had Severus been less ambitious for establishing a dynasty.

Active valour may often be the present of nature; but such patient diligence can be the fruit only of habit and discipline.

In this quote, Gibbon is praising the work of the Roman Legions who laid out their camps with the precision of cities and did not shirk from digging well-made ditches and building careful palisades. He is saying only well-trained troops can achieve this: just being naturally active cannot.

All three of these quotes refer to a specific situation on which Gibbon is commenting. However, all three contain general wisdom that is applicable to many situations, which is why they have been remembered.