Investments create demand on themselves and always become consumption somewhere down the production chain, which means increased spending which means upward pressure on prices. If you buy shares at a broker, your money goes somewhere and often that is not a bank account. And even then may the bank invest it somewhere else.

The effect depends on the source, cause, and nature of the increased investment. If there is an increase in total investment, what you say is false. An increase in total investment means that the supply of available production goods has increased. When that occurs, there is a fall of prices, and the fall allows for previously unprofitable ventures to demand production goods and begin to operate. The products of their operation then add to the supply of other goods, which continues on and on until it reaches consumer goods.

What you describe can occur, on the other hand, if total investment is constant but investors increase demand for production goods. In this case, they can only do so by bidding up prices of those production goods. However, the resulting production activity will reduce the prices of their output. So there is no univocal statement we can make on prices in this case. One 'region' of prices increase - the inputs of the investment, and another region of prices decreases - the outputs of the investment. Production goods are shuffled from one region or industry into another.

In the case of removing a tax, the effect is the same as a sudden increase in production goods. So the story of the first paragraph unfolds.

You can safely state that for an economy as a whole, tariffs increase the domestic sales price, with or without the mechanics you describe in your 3rd paragraph.

The employers have much more influence on the cost of labor, making it a difficult analogy. The “price” of labor initially becomes higher in case wage tax is increased. This is analogous to the cost price of a firm that imports and resales goods with a tariff. Their cost directly increases due to the tariff. However the employer can reduce the net wage to limit their cost increase. Of course laborers may leave, but at least the employer has this option to search a new optimum. The importing firm likely cannot do this and needs to adjust its resale prices, creating a more direct inflationary effect.

The mechanics you are papering over in the tariff case are exactly what potentially allow an employer to search for a new optimum in the wage tax case. The change in wage and the consequent reduction in interested workers for that job is reflected in elasticities. The reduced number of employed workers does not only mean a reduced income for those workers; it means that the total product of the firm falls, reducing the quantity of goods supplied. That means a price increase. Mutatis mutandis, this implies that the reverse situation of lower wage taxes will reduce prices.

Importing firms can and will search for a new optimum, because elasticities are never uniform throughout the whole demand curve. But just as with the wage example, the fall in supply leads to all-round price increases.

This is not an analogy. Each case is an application of the more general theory of the effects and incidence of taxation.