Many who are in Christian leadership positions use Ps.105:15 to tell others, "Do not touch My anointed ones," meaning no one should question them or talk about their faults. Some go to the extent of threatening calamity to those who dare to oppose them.
What is happening at ground level can be of two types. Some people oppose the authority which God has given to the leaders because of their own rebellious attitudes. But some others don't dare to raise any issues or question misuse of authority because of fear of God's judgment. We need to see these as two different dangers.
Those who oppose or resist the authority which God has placed on the leaders face severe judgment from God (Ro.13:1,2). Miriam and Aaron questioned the authority of Moses because they thought they also deserved some of that authority, probably because they were his older siblings (Nu.12:1,2). A cousin of Moses, Korah, gathered about 250 people against Moses saying that he was exalting himself over them (Nu.16:3). In both these cases Moses had done nothing wrong, and this could also be seen by the way he responded to these accusations. God judged Miriam with leprosy and He opened the earth miraculously and buried Korah and his group alive for their affront. These are warnings to us not to resist or rebel against authority, especially thinking too much of ourselves or seeking something for ourselves.
But there are also many cases where people in positions of authority misuse it and lord it over others, fleece or even abuse them. Are we supposed to ignore such things, not to talk about them with others and to take things as if coming from the authority of God? It is because many are afraid to even question such things that these so-called leaders 'flourish' without opposition and damage many lives and dishonour God. Does God want us to silently let that happen?
"Do not receive an accusation against an elder except on the basis of two or three witnesses. Those who continue in sin, rebuke in the presence of all, so that the rest also will be fearful of sinning" (1Ti.5:19,20). For this to happen, some people should have been talking about it! They discussed matters among themselves discretely and finally took some action. Obviously they were not being rebellious or scheming against the leaders! But they were concerned for the leaders and for God's name. This needed humility, discernment and discretion. But that wouldn't have happened if they were cowed down by a 'simplistic' understanding about resisting authority.
What God looks at first of all is the motive of our heart whatever we do (1Co.4:5). A 'junior apostle' Paul had the 'audacity' to question the senior apostle Peter (Ga.2:11), and Peter took it well. Jesus allowed Peter to rebuke Him (Mt.16:22), even though He corrected Peter afterwards. But leaders who feel insecure and threatened warn people with Ps.105:15 (where, actually, 'the anointed' refers to the people and not leaders)!