I don’t think there is an explicit definitive answer to this question. However, there are several points to consider that may help address it.
First, the premise of the question is unsubstantiated. How do you know that Dumbledore didn’t investigate? Just because it’s not mentioned “on camera”, doesn’t mean it didn’t happen.
But if we insist on assuming that Dumbledore did not investigate, we should consider how much Dumbledore knew, and when he knew it. Note that Dumbledore doesn’t mention anything to Harry about the Riddle house. If he was trying to impress upon Harry the connection between Frank and Voldemort, wouldn’t you expect him to mention that Frank was actually the caretaker of the Riddle house, not merely a resident of the village where Voldemort’s father grew up? It is possible, then, that Dumbledore was unaware of this additional connection, and would thus have no reason to suspect that anything happened at the Riddle house specifically.
Additionally, we don’t know when Dumbledore found out about Frank’s disappearance. In many (most?) jurisdictions a person will not be considered missing unless a certain amount of time has elapsed. Frank being an old guy living alone with apparently not much of a social life would likely contribute to additional delays in people noticing that he was missing. And even once it was determined that he was missing, we don’t know how long it was until it was published in the newspaper. And we further don’t know how much time elapsed before Dumbledore read it in the newspaper.
When Dumbledore says that he reads the Muggle newspaper, does he mean that he reads every issue of every Muggle newspaper? Probably not, as that would use up a good portion of his time, and he has some other rather important duties in his life. It is possible that Dumbledore does not read the Muggle newspaper religiously, but merely means that he doesn’t object to reading it (as do his colleagues) and therefore reads it on occasion when the opportunity presents itself. This may be similar to what we see in Chapter Four of Half-Blood Prince where Dumbledore apparently catches up on his magazine reading when presented with a stash in a Muggle house:
“Oh, there you are, Albus,” he said. “You’ve been a very long time. Upset stomach?”
“No, I was merely reading the Muggle magazines,” said Dumbledore. “I do love knitting patterns.
(Of course, in that situation Dumbledore was deliberately giving Slughorn time alone with Harry, so it is possible that he was merely faking interest in the magazines.)
The point of all this is that it is possible that Dumbledore did not became aware of Frank’s disappearance until weeks, even months, after it happened. By that time he might not have expected there to be anything left to investigate. We know from Chapter One of Goblet of Fire that Voldemort was not planning on remaining in the house for much longer, even before Frank showed up:
“My Lord, may I ask how long we are going to stay here?”
“A week,” said the cold voice. “Perhaps longer. The place is moderately comfortable, and the plan cannot proceed yet. It would be foolish to act before the Quidditch World Cup is over.”
And after killing Frank they may have cut their stay even shorter. So it is quite possible that all evidence was long gone by the time Dumbledore could have conducted any investigation, and he thus considered it a futile endeavor.
It also seems pretty consistent with Dumbledore’s general personality to not investigate this. In just about every book in the series there is something going on that Dumbledore could have invested more in invested more in investigating and/or thwarting. However, he repeatedly chooses to let things play out, and interferes as little as possible (or even helps things along!), whether it’s the Philosopher’s Stone, the Chamber of Secrets, the Triwizard Tournament, Malfoy’s escapades, etc.
In sum, we can address the question from several angles:
- Perhaps he did investigate.
- Perhaps he was unaware of the connection to the Riddle house.
- Perhaps he became aware too late.
- Perhaps he did not want to disrupt the natural course of events.