Hard to say definitively without seeing the article in question, but often a newspaper headline will use the "to" phrasing as an abbreviation for "set to" or "plans to" or something similar. This is not something specific to news, per se, but rather indicates a slightly different meaning than will.
In this particular case, I'd guess that the headline is a shorter version of "Donald Trump plans to run for president next month." It's likely that if you compare the date of publication against the date Donald Trump either a) publicly announced his intention to run for president or b) filed the official paperwork necessary to run for president, you'll find that the article was published before either of those things happened.
Likely, the reporter writing the article had a source telling them that Donald Trump was planning to run for president, but Trump did not tell the reporter himself that he was going to run, nor did Trump make any sort of formal announcement that he would run, so the reporter did not want to say "will" because "will" would imply a certainty that the reporter did not have.
Alternatively, if the reporter did know for certain that Trump was planning to run for president, the use of "to" instead of "will" could simply be a style choice meant to keep the headline in present tense instead of future tense. Even with confirmation that something is officially set to happen, newspapers are still typically hesitant to use a "will happen" type of phrasing because you never know what could happen in the future. They simply report what the case is at the time of publication.