A company typically issues warrants* to investors & institutions participating in a new share or bond issue. The warrant is a "kicker" to sweeten the deal by granting participants the right, but not the obligation, to acquire stock in the company at a set price, by a given date. (On the surface, warrants are similar to call options, but different in other respects. Perhaps another question.)
Usually, the warrants are priced out of the money, but with ample time before expiration to provide for the possibility that good company performance will raise the underlying share price above the exercise price, making the warrants worth exercising.
Since warrants tend to be issued as one part of a deal, the stock price at the time of the deal will primarily be affected by the principal financial instruments involved. The market will weigh the deal's pricing, reasons for fundraising, and likelihood for positive or negative outcomes. For instance:
If new shares are issued, are they at a steep discount to recent share prices? Shareholders don't like to see new shares issued at a discount, yet sometimes companies do it because they're desperate and nobody would pay a higher price. The market may drive down the share price.
If new shares are issued, are they at a price consistent with recent share prices? Maybe no impact. On the other hand, were the funds raised, say, to pay down high-interest debt, or fund an eagerly-anticipated expansion to fill unmet product demand? If the reason for raising money has a good probability of growing earnings, the market may react positively, else yawn.
If new bonds are issued, are they at an interest rate higher than expected? Shareholders seeing higher interest costs ahead, or who interpret the higher rate as lower confidence from creditors may drive the share price down. Or, if shareholders had been worried a money-losing company might be unable to raise money at all, i.e. possible bankruptcy, then the shares could have initially been distressed, and success even at the high interest rate might be interpreted as a positive sign.
If new bonds are issued, are they at an interest rate lower than expected? As with the share issuance case, shareholders could react positively because the funds could, say, retire existing higher-interest debt, or fuel growth.
Those aside, let's consider just the warrants:
First, at the time of the deal, yes, the warrants can impact the share price:
If the exercise price is set too close to the current share price, shareholders might assign a higher probability that the warrants will ultimately be exercised, resulting in dilution of earnings per share. Everything else being equal, high probability of future earnings dilution (and to what extent) is likely to have a negative impact on share price — however, you might not see any drop due to the warrants alone if the deal's primary instruments (above) gave the shares sufficient lift.
If the exercise price is sufficiently higher than the current share price, shareholders might assign a lower probability for exercise, i.e. less likely resulting in dilution. Everything else being equal, this is unlikely to have much of an impact on the share price — however, you might still see a change in the share price due to the deal's primary instruments (above).
Next, at the time of exercise, yes, the warrants can also impact the share price. Recognize that the warrants are being exercised because the company share price has exceeded the warrant exercise price. These are two direct consequences of the exercise:
The actual increase in the number of shares outstanding, and the resulting dilution of earnings, are weighed more heavily by shareholders than the mere possibility of such. Yet, it's likely this will have been realized gradually by the market as the share price approached the warrant exercise price.
The new shares are being issued at a discount to the current market price. If the price paid by warrant holders to acquire shares is significantly below the company's net assets per share, then shareholders may also see this reduction in net assets per share as a reason to reprice the shares in the market.
There may be other reasons yet, but I hope I convinced that the market is (a) complex, and (b) constantly reassessing what a share might be worth based on a variety of changing factors & expectations. It's simplistic to assume the impact of a warrant issue will occur all up front, or all upon exercise.
* n.b. I am referring to "warrants" in the traditional sense as used in the question, i.e. "financing warrants", typically issued by the company itself and, when exercised, resulting in issuance of new shares. Contrast with "Australian warrants" which are issued by 3rd party institutional market participants, like a pure derivative product or exchange traded option – i.e. no new shares are created. (Thanks, @Victor, for mentioning the distinction and supplying the link.)